Shrinkage, sharing power, and logistic regression.
Peter Ralph
19 November 2019 – Advanced Biological Statistics
Last time: baseball
The model
\[\begin{aligned} Z_i &\sim \Binom(N_i, \theta_i) \\ \theta_i &\sim \Beta(\mu_{p_i} \kappa_{p_i}, (1-\mu_{p_i})\kappa_{p_i}) \\ \mu &\sim \Beta(1, 1) \\ \kappa_p &\sim \Gam(0.1, 0.1) . \end{aligned}\]
pos_model <- "
data {
int N; // number of players
int hits[N];
int at_bats[N];
int npos; // number of positions
int position[N];
}
parameters {
real<lower=0, upper=1> theta[N];
real<lower=0, upper=1> mu[npos];
real<lower=0> kappa[npos];
}
model {
real alpha;
real beta;
hits ~ binomial(at_bats, theta);
for (i in 1:N) {
alpha = mu[position[i]] * kappa[position[i]];
beta = (1 - mu[position[i]]) * kappa[position[i]];
theta[i] ~ beta(alpha, beta);
}
mu ~ beta(1,1);
kappa ~ gamma(0.1,0.1);
}"batting <- read.csv("BattingAverage.csv", header=TRUE)
baseball_model <- stan_model(model_code=pos_model)system.time(pos_fit <- sampling(baseball_model,
chains=3, iter=1000,
control=list(max_treedepth=13),
data=list(N=nrow(batting),
hits=batting$Hits,
at_bats=batting$AtBats,
npos=nlevels(batting$PriPos),
position=as.numeric(batting$PriPos))))## user system elapsed
## 337.477 0.471 253.700The results
## Warning: Removed 18 rows containing missing values (geom_bar).
With labels: position means \(\mu_p\):
Position “concentrations”, \(\kappa_p\):
How well are we doing?
Simulation
Wouldn’t it be nice if we knew the truth?
Discuss:
Write down (on whiteboards) a procedure to simulate data that looks like the baseball data.
batting$post_mean <- colMeans(param_samples$theta)
batting$fake_hits <- rbinom(nrow(batting),
size=batting$AtBats,
prob=batting$post_mean)Option 1: By simulating draws from the posterior mean on \(\theta\), we can check that our model is accurately describing the data. Here is the real data against data simulated under the posterior mean \(\theta\) values. They look similar, which is good.
But if we want to know if we’re accurately estimating \(\mu\) and \(\kappa\), then we have to start with them, and simulate \(\theta\).
post_mean_mu <- colMeans(param_samples$mu)
post_mean_kappa <- colMeans(param_samples$kappa)
names(post_mean_mu) <- names(post_mean_kappa) <- levels(batting$PriPos)
batting$sim_theta <- rbeta(nrow(batting),
shape1=post_mean_mu[batting$PriPos] *
post_mean_kappa[batting$PriPos],
shape2=(1-post_mean_mu[batting$PriPos]) *
post_mean_kappa[batting$PriPos])
batting$sim_theta_hits <- rbinom(nrow(batting),
size=batting$AtBats,
prob=batting$sim_theta)Fit the model to the simulated data:
Can we estimate \(\mu\) and \(\kappa\)?
sim_samples <- rstan::extract(sim_fit)
layout(t(1:2))
boxplot(sim_samples$mu, main="posterior distribution of mu")
points(1:9, post_mean_mu, col='red', pch=20, cex=2)
boxplot(sim_samples$kappa, main="posterior distribution of kappa")
points(1:9, post_mean_kappa, col='red', pch=20, cex=2)
General questions with simulated data
- Does my statistical inference method work?
Do the credible intervals contain the true value?
(i.e., Is the method “well-calibrated”?)
They usually should.
How wide are credible intervals, typically?
This is (one kind of) statistical power.
Posterior predictive sampling
How to choose parameter values for simulation?
It depends, but one good choice is from the posterior distribution - then you can see how the model behaves with data close to the real data.
This is known as posterior predictive sampling.
Sharing power // Shrinkage
Example
Suppose that I have a large pot of coins that are all similar to each other. I flip each one ten times, and record the number of Heads. What is each coin’s probability of Heads?
Treated separately, we would be very uncertain about each coin.
Together, they should tell us very accurately what are likely values of \(\theta\).
This information can improve the estimate of each separate \(\theta\).
The more similar the coins are, the more information we gain.
By shrinking each estimate towards their common mean, we hope to gain power.
Shrinkage and baseball
Some players were at bat very few times. How does the information about their position affect our inference about their batting averages?
batting$post_med <- colMedians(param_samples$theta)
batting$post_Q1 <- colQuantiles(param_samples$theta, probs=0.25)
batting$post_Q3 <- colQuantiles(param_samples$theta, probs=0.75)
pos_means <- colMeans(param_samples$mu)
names(pos_means) <- levels(batting$PriPos)
pos_means## 1st Base 2nd Base 3rd Base Catcher Center Field
## 0.2526803 0.2490924 0.2560258 0.2390829 0.2577250
## Left Field Pitcher Right Field Shortstop
## 0.2493405 0.1288152 0.2582038 0.2477235Pitchers had posterior mean \(\mu\) of 0.1288152
with(subset(batting[order(batting$post_med),], PriPos=="Pitcher"), {
plot(Hits / AtBats, main="Pitchers", xaxt='n', ylim=c(0, 0.4),
xlab='player', ylab="posterior median theta");
segments(x0=seq_along(Hits), y0=post_Q1, y1=post_Q3,
col=adjustcolor('red',0.5));
points(post_med, pch=20) })
abline(h=pos_means["Pitcher"], lwd=2, col=adjustcolor("blue", 0.5))
Catchers had posterior mean \(\mu\) of 0.2390829
with(subset(batting[order(batting$post_med),], PriPos=="Catcher"), {
plot(Hits / AtBats, main="Catchers", xaxt='n', ylim=c(0, 0.4),
xlab='player', ylab="posterior median theta");
segments(x0=seq_along(Hits), y0=post_Q1, y1=post_Q3,
col=adjustcolor('red',0.5));
points(post_med, pch=20) })
abline(h=pos_means["Catcher"], lwd=2, col=adjustcolor("blue", 0.5))
Is shrinkage here a good idea?
With simulated data, compare median absolute error for
posterior mean \(\theta_i\)
empirical batting average
How’d we do? Let’s look at the true values of \(\theta\) (we know because we generated them) against the posterior means. Red lines are 95% credible intervals.
post_sim_theta <- colMeans(sim_samples$theta)
plot(batting$sim_theta, post_sim_theta, type='n', xlab="true theta", ylab="posterior mean theta", ylim=c(0,0.4))
segments(x0=batting$sim_theta, col='red',
y0=colQuantiles(sim_samples$theta, probs=0.05),
y1=colQuantiles(sim_samples$theta, probs=0.95))
points(batting$sim_theta, post_sim_theta, pch=20)
abline(0, 1, lwd=3, col=adjustcolor('blue', 0.5))
Did we do better?
The mean absolute difference between the truth and
- posterior mean: 0.0254728
- empirical batting average (hits/at bats): 0.0893981
Using information about groups reduced our error by a factor of 4!
Stochastic minute
Exponential, and Gamma
If \(T \sim \Exp(\text{rate}=\lambda)\), then
\[\begin{aligned} \P\{ T \in dt \} = \lambda e^{-\lambda t} dt . \end{aligned}\]
\(T\) can be any nonnegative real number.
\(T\) is memoryless: \[\begin{aligned} \P\{ T > x + y \given T > x \} = \P\{ T > y \} . \end{aligned}\]
A machine produces \(n\) widgets per second; each widget has probability \(\lambda/n\) of being broken. The time until the first broken widget appears (in seconds) is approximately \(\sim \Exp(\lambda)\).
If \(S \sim \Gam(\text{shape}=\alpha, \text{rate}=\lambda)\), then
\[\begin{aligned} \P\{ S \in dt \} = \frac{\alpha^\lambda}{\Gamma(\alpha)} t^{\alpha - 1} e^{-\lambda t} dt . \end{aligned}\]
If \(T_1, \ldots, T_k\) are independent \(\Exp(\lambda)\), then \(S = T_1 + \cdots + T_k\) is \(\Gam(k, \lambda)\).
A machine produces \(n\) widgets per second; each widget has probability \(\lambda/n\) of being broken. The time until the \(k^\text{th}\) broken widget appears (in seconds) is approximately \(\sim \Gam(k, \lambda)\).
Logistic regression, a.k.a. metric predictors
Motivation
There’s a good bit more information available about the baseball players.
## Player PriPos Hits AtBats PlayerNumber PriPosNumber
## 1 Aaron Cunningham Left Field 17 97 198 7
## 2 Aaron Harang Pitcher 4 56 360 1
## 3 Aaron Hill 2nd Base 184 609 394 4
## 4 Aaron Laffey Pitcher 0 1 482 1
## 5 Aaron Loup Pitcher 0 1 522 1
## 6 Adam Dunn 1st Base 110 539 242 3
## playerID weight height bats birthYear debut finalGame
## 1 cunniaa01 215 71 R 1986 2008-08-31 2012-07-24
## 2 haranaa01 260 79 R 1978 2002-05-25 2015-10-03
## 3 hillaa01 200 71 R 1982 2005-05-20 2016-10-01
## 4 laffeaa01 190 71 L 1985 2007-08-04 2015-07-31
## 5 loupaa01 210 71 L 1987 2012-07-14 2016-09-30
## 6 dunnad01 285 78 L 1979 2001-07-20 2014-09-28Is batting average predicted by weight and height?
Each player has their own batting avg, \(\theta_i\).
Batting averages tend to be different for different positions.
After accounting for position (\(p_i\)), \(\theta\) varies with height (\(h_i\)) and/or weight (\(w_i\)).
\[\begin{aligned} Z_i &\sim \Binom(N_i, \theta_i) \\ \theta_i &\sim \Beta(\mu_i \kappa_{p_i}, (1-\mu_i) \kappa_{p_i}) \\ \mu_i &= \logistic(\gamma_{p_i} + \alpha_h h_i + \alpha_w w_i) \\ \kappa_p &\sim \Gam(0.1, 0.1) \\ \gamma_{p} &\sim \Normal(0, \sigma_\gamma) \\ \alpha_h &\sim \Normal(0, \sigma_h) \\ \alpha_w &\sim \Normal(0, \sigma_w) \\ \sigma_\gamma, \sigma_h, \sigma_w &\sim \Normal^+(0, 20) \end{aligned}\]
Stan
logistic_model <- "
data {
int N; // number of players
int hits[N];
int at_bats[N];
int npos; // number of positions
int position[N];
vector[N] height;
vector[N] weight;
}
parameters {
real<lower=0, upper=1> theta[N];
vector[npos] gamma;
real alpha_w;
real alpha_h;
vector<lower=0>[npos] kappa;
}
model {
vector[N] mu;
mu = inv_logit(gamma[position] + alpha_h * height + alpha_w * weight);
theta ~ beta(mu .* kappa[position], (1-mu) .* kappa[position]);
hits ~ binomial(at_bats, theta);
kappa ~ gamma(0.1, 0.1);
gamma ~ normal(0, 5);
alpha_h ~ normal(0, 5);
alpha_w ~ normal(0, 5);
} "\[\begin{aligned} Z_i &\sim \Binom(N_i, \theta_i) \\ \theta_i &\sim \Beta(\mu_i \kappa_{p_i}, (1-\mu_i) \kappa_{p_i}) \\ \mu_i &= \logistic(\gamma_{p_i} + \alpha_h h_i + \alpha_w w_i) \\ \kappa_p &\sim \Gam(0.1, 0.1) \\ \gamma_{p} &\sim \Normal(0, 5) \\ \alpha_h &\sim \Normal(0, 5) \\ \alpha_w &\sim \Normal(0, 5) \end{aligned}\]
New features
logistic_model <- "
data {
int N; // number of players
int hits[N];
int at_bats[N];
int npos; // number of positions
int position[N];
vector[N] height;
vector[N] weight;
}
parameters {
real<lower=0, upper=1> theta[N];
vector[npos] gamma;
real alpha_w;
real alpha_h;
vector<lower=0>[npos] kappa;
}
model {
vector[N] mu;
mu = inv_logit(gamma[position] + alpha_h * height + alpha_w * weight);
theta ~ beta(mu .* kappa[position], (1-mu) .* kappa[position]);
hits ~ binomial(at_bats, theta);
kappa ~ gamma(0.1, 0.1);
gamma ~ normal(0, 5);
alpha_h ~ normal(0, 5);
alpha_w ~ normal(0, 5);
} "vector[N] height;instead ofreal height[N];: to allowalpha_h * height.we will center and scale height and weight before passing to Stan
x .* yfor component-wise multiplication
data {
int N; // number of players
int hits[N];
int at_bats[N];
int npos; // number of positions
int position[N];
vector[N] height;
vector[N] weight;
}
parameters {
real<lower=0, upper=1> theta[N];
vector[npos] gamma;
real alpha_w;
real alpha_h;
vector<lower=0>[npos] kappa;
}
model {
vector[N] mu;
mu = inv_logit(gamma[position] + alpha_h * height + alpha_w * weight);
theta ~ beta(mu .* kappa[position], (1-mu) .* kappa[position]);
hits ~ binomial(at_bats, theta);
kappa ~ gamma(0.1, 0.1);
gamma ~ normal(0, 5);
alpha_h ~ normal(0, 5);
alpha_w ~ normal(0, 5);
} scaled_height <- (batplus$height - mean(batplus$height))/sd(batplus$height)
scaled_weight <- (batplus$weight - mean(batplus$weight))/sd(batplus$weight)
system.time(logistic_fit <- stan(model_code=logistic_model, chains=3, iter=1000,
control=list(adapt_delta=0.9, max_treedepth=12),
data=list(N=nrow(batplus),
hits=batplus$Hits,
at_bats=batplus$AtBats,
npos=nlevels(batplus$PriPos),
position=as.numeric(batplus$PriPos),
height=scaled_height,
weight=scaled_weight) ))## user system elapsed
## 349.103 3.311 234.642## mean se_mean sd 2.5%
## theta[1] 2.145500e-01 0.0005779793 0.02932482 1.593682e-01
## theta[2] 1.006387e-01 0.0005855878 0.02971730 5.203308e-02
## theta[3] 2.952353e-01 0.0002946572 0.01717832 2.631069e-01
## theta[4] 1.260534e-01 0.0009424901 0.04849544 4.261734e-02
## theta[5] 1.275896e-01 0.0008539137 0.04622264 5.069259e-02
## theta[6] 2.138239e-01 0.0002752581 0.01663715 1.815804e-01
## theta[7] 2.833089e-01 0.0003267569 0.01618581 2.528106e-01
## theta[8] 2.593683e-01 0.0004990926 0.02785333 2.071012e-01
## theta[9] 2.677935e-01 0.0002903689 0.01699189 2.349905e-01
## theta[10] 2.540948e-01 0.0004175393 0.02148285 2.141433e-01
## theta[11] 2.301409e-01 0.0008895427 0.04261473 1.502910e-01
## theta[12] 1.252804e-01 0.0007377251 0.04206873 5.302340e-02
## theta[13] 2.355161e-01 0.0005978733 0.03103126 1.741003e-01
## theta[14] 1.256627e-01 0.0005887235 0.02991761 7.227218e-02
## theta[15] 2.541204e-01 0.0005462230 0.02907560 2.002613e-01
## theta[16] 3.128295e-01 0.0003468668 0.01811826 2.788633e-01
## theta[17] 2.251309e-01 0.0006361174 0.03425126 1.581913e-01
## theta[18] 2.918966e-01 0.0002778286 0.01746653 2.583556e-01
## theta[19] 2.454895e-01 0.0007195728 0.03553543 1.788194e-01
## theta[20] 2.805433e-01 0.0002964139 0.01780158 2.466547e-01
## theta[21] 2.535251e-01 0.0003510053 0.01873934 2.182989e-01
## theta[22] 2.462706e-01 0.0006942018 0.03526476 1.784789e-01
## theta[23] 2.870388e-01 0.0002871568 0.01642285 2.568942e-01
## theta[24] 2.765417e-01 0.0002834955 0.01708938 2.434321e-01
## theta[25] 2.414787e-01 0.0003487306 0.01927052 2.038662e-01
## theta[26] 2.353474e-01 0.0006510681 0.03697088 1.640661e-01
## theta[27] 2.884944e-01 0.0002882870 0.01665930 2.570944e-01
## theta[28] 1.220019e-01 0.0009455966 0.04523748 5.047242e-02
## theta[29] 2.388824e-01 0.0005550809 0.02992790 1.822866e-01
## theta[30] 2.399783e-01 0.0003454758 0.02075186 2.000419e-01
## theta[31] 2.964206e-01 0.0002772313 0.01709152 2.648006e-01
## theta[32] 2.697496e-01 0.0003324702 0.01858642 2.334414e-01
## theta[33] 1.463483e-01 0.0007212393 0.04003268 7.926643e-02
## theta[34] 2.622054e-01 0.0002626519 0.01659561 2.306560e-01
## theta[35] 2.422328e-01 0.0004751894 0.02259819 2.004356e-01
## theta[36] 2.420279e-01 0.0003695997 0.02127655 2.024598e-01
## theta[37] 1.410335e-01 0.0009126049 0.04889367 6.236259e-02
## theta[38] 2.595456e-01 0.0003441256 0.01609152 2.282160e-01
## theta[39] 1.211741e-01 0.0008507381 0.04352680 4.958072e-02
## theta[40] 2.254632e-01 0.0008830824 0.04177440 1.452957e-01
## theta[41] 2.969712e-01 0.0003055442 0.01892233 2.612358e-01
## theta[42] 2.795464e-01 0.0003516759 0.01758941 2.456152e-01
## theta[43] 2.745576e-01 0.0005462487 0.02927104 2.171541e-01
## theta[44] 2.358030e-01 0.0003966881 0.01950791 1.980122e-01
## theta[45] 1.315614e-01 0.0007512199 0.04460976 5.584001e-02
## theta[46] 3.173937e-01 0.0003551549 0.01732124 2.843139e-01
## theta[47] 2.752675e-01 0.0007672610 0.04399289 1.940284e-01
## theta[48] 1.198761e-01 0.0007595353 0.03985946 5.372125e-02
## theta[49] 2.141447e-01 0.0003429838 0.02237976 1.724104e-01
## theta[50] 3.029753e-01 0.0003714826 0.02189539 2.607143e-01
## theta[51] 2.235166e-01 0.0004601774 0.02906924 1.679899e-01
## theta[52] 1.390218e-01 0.0009053833 0.04630519 6.186642e-02
## theta[53] 1.512535e-01 0.0011437874 0.05163740 6.700166e-02
## theta[54] 2.840219e-01 0.0002854407 0.01684076 2.527143e-01
## theta[55] 1.189585e-01 0.0006952370 0.03661201 5.507241e-02
## theta[56] 1.279335e-01 0.0006610448 0.03783374 6.231169e-02
## theta[57] 2.356423e-01 0.0005049255 0.02540414 1.866452e-01
## theta[58] 2.057146e-01 0.0006778799 0.03661415 1.394553e-01
## theta[59] 2.768430e-01 0.0004179071 0.02213817 2.340490e-01
## theta[60] 1.231400e-01 0.0008357619 0.04519562 4.936545e-02
## theta[61] 2.932269e-01 0.0003483945 0.01831330 2.580207e-01
## theta[62] 1.130078e-01 0.0007806604 0.04137872 4.432528e-02
## theta[63] 2.678466e-01 0.0003138536 0.01726272 2.335964e-01
## theta[64] 2.267506e-01 0.0005659973 0.03081056 1.691393e-01
## theta[65] 2.942472e-01 0.0002696560 0.01786884 2.591023e-01
## theta[66] 2.482215e-01 0.0005202538 0.02757780 1.953908e-01
## theta[67] 2.795210e-01 0.0006523930 0.03601468 2.114380e-01
## theta[68] 1.293785e-01 0.0008471966 0.04998868 4.983809e-02
## theta[69] 1.015309e-01 0.0005810056 0.02964064 4.978839e-02
## theta[70] 1.288167e-01 0.0010053265 0.04662206 5.347122e-02
## theta[71] 2.457908e-01 0.0004379617 0.02545935 1.982449e-01
## theta[72] 2.867009e-01 0.0002931163 0.01905086 2.503189e-01
## theta[73] 1.006322e-01 0.0007681301 0.03822340 3.768002e-02
## theta[74] 2.677368e-01 0.0002739098 0.01698496 2.342830e-01
## theta[75] 3.047994e-01 0.0003065367 0.01755585 2.700219e-01
## theta[76] 1.261247e-01 0.0010886918 0.04655243 4.911701e-02
## theta[77] 2.240739e-01 0.0006627528 0.03833681 1.483713e-01
## theta[78] 2.209284e-01 0.0007660789 0.04139666 1.462713e-01
## theta[79] 2.310713e-01 0.0008063982 0.04009743 1.586821e-01
## theta[80] 2.451550e-01 0.0004414278 0.02419940 1.987738e-01
## theta[81] 2.205431e-01 0.0005061940 0.02727173 1.659928e-01
## theta[82] 1.283953e-01 0.0009609605 0.04685166 5.179558e-02
## theta[83] 1.281793e-01 0.0008475721 0.04730908 5.097306e-02
## theta[84] 1.268587e-01 0.0008796083 0.04814407 4.800774e-02
## theta[85] 1.426179e-01 0.0009137986 0.04613320 6.474358e-02
## theta[86] 1.322829e-01 0.0008291747 0.04879226 5.228612e-02
## theta[87] 1.363545e-01 0.0009326539 0.04438622 6.172459e-02
## theta[88] 2.277515e-01 0.0007423547 0.03778234 1.549360e-01
## theta[89] 2.290870e-01 0.0006321464 0.03069941 1.695159e-01
## theta[90] 1.496301e-01 0.0007215057 0.03975183 8.057962e-02
## theta[91] 2.705004e-01 0.0003491137 0.02050104 2.302107e-01
## theta[92] 2.480946e-01 0.0002964102 0.01816552 2.138871e-01
## theta[93] 1.399360e-01 0.0009662876 0.04754797 6.309035e-02
## theta[94] 1.256734e-01 0.0008409116 0.04552127 5.023225e-02
## theta[95] 2.393426e-01 0.0008164256 0.04071062 1.625726e-01
## theta[96] 2.162851e-01 0.0006986278 0.03446686 1.516542e-01
## theta[97] 2.260366e-01 0.0003749624 0.02149336 1.853320e-01
## theta[98] 2.565850e-01 0.0007430360 0.03804858 1.841898e-01
## theta[99] 2.797796e-01 0.0004326345 0.02262473 2.361048e-01
## theta[100] 2.762879e-01 0.0002839514 0.01729488 2.437909e-01
## theta[101] 2.614040e-01 0.0006031869 0.03547793 1.950552e-01
## theta[102] 2.385227e-01 0.0004365910 0.02325803 1.949788e-01
## theta[103] 2.031433e-01 0.0003072418 0.01808164 1.704010e-01
## theta[104] 2.436818e-01 0.0003763413 0.01914067 2.078174e-01
## theta[105] 2.144770e-01 0.0004246796 0.02339922 1.680733e-01
## theta[106] 2.141645e-01 0.0005929390 0.02941541 1.575346e-01
## theta[107] 2.544934e-01 0.0011130972 0.04617076 1.685340e-01
## theta[108] 1.267387e-01 0.0008743834 0.04595058 5.006145e-02
## theta[109] 2.643913e-01 0.0007036442 0.03737189 1.960703e-01
## theta[110] 2.166846e-01 0.0004507483 0.02700690 1.666349e-01
## theta[111] 2.118392e-01 0.0005211641 0.02830469 1.595850e-01
## theta[112] 2.704698e-01 0.0003411833 0.01778950 2.365732e-01
## theta[113] 2.297308e-01 0.0004665930 0.02901849 1.762938e-01
## theta[114] 2.554897e-01 0.0004485118 0.02462024 2.103905e-01
## theta[115] 2.285475e-01 0.0006044937 0.03025059 1.702306e-01
## theta[116] 2.157893e-01 0.0003936418 0.01958533 1.774154e-01
## theta[117] 2.378530e-01 0.0004527201 0.02225222 1.973459e-01
## theta[118] 1.248323e-01 0.0009416983 0.04934010 4.460446e-02
## theta[119] 1.353789e-01 0.0010129102 0.04604627 6.221378e-02
## theta[120] 2.318852e-01 0.0002959396 0.01717177 1.993187e-01
## theta[121] 2.220568e-01 0.0005835889 0.03166927 1.664617e-01
## theta[122] 1.245138e-01 0.0008880967 0.04573616 4.785133e-02
## theta[123] 2.316222e-01 0.0005346042 0.03043244 1.736873e-01
## theta[124] 2.651177e-01 0.0006674468 0.03377198 2.025899e-01
## theta[125] 1.363134e-01 0.0006794783 0.03371314 7.718086e-02
## theta[126] 1.916944e-01 0.0005614366 0.02926282 1.366497e-01
## theta[127] 1.386285e-01 0.0008035015 0.04667098 6.068717e-02
## theta[128] 1.349649e-01 0.0008975205 0.04560827 5.921357e-02
## theta[129] 2.369053e-01 0.0007998861 0.04188165 1.610708e-01
## theta[130] 2.378897e-01 0.0007728074 0.04280332 1.585211e-01
## theta[131] 2.575920e-01 0.0003690515 0.01995515 2.195538e-01
## theta[132] 2.107348e-01 0.0003847160 0.02263931 1.686191e-01
## theta[133] 2.686318e-01 0.0003065711 0.01679608 2.362626e-01
## theta[134] 1.150407e-01 0.0006285087 0.03039739 6.117863e-02
## theta[135] 1.235513e-01 0.0008651427 0.04569515 5.173109e-02
## theta[136] 3.216548e-01 0.0003798978 0.01930765 2.853599e-01
## theta[137] 2.446747e-01 0.0003004237 0.01752062 2.105518e-01
## theta[138] 2.672723e-01 0.0004976355 0.02996013 2.120495e-01
## theta[139] 1.264969e-01 0.0008924316 0.04562596 4.921105e-02
## theta[140] 2.674227e-01 0.0003054010 0.01850560 2.314390e-01
## theta[141] 2.543318e-01 0.0005968485 0.03671041 1.869009e-01
## theta[142] 2.603863e-01 0.0003420751 0.01887453 2.256267e-01
## theta[143] 2.938429e-01 0.0002792775 0.01822682 2.586870e-01
## theta[144] 2.634031e-01 0.0003302559 0.01710324 2.310693e-01
## theta[145] 2.350720e-01 0.0005002953 0.03314273 1.694597e-01
## theta[146] 2.070566e-01 0.0002872298 0.01611063 1.763679e-01
## theta[147] 2.581314e-01 0.0004125782 0.02272291 2.148104e-01
## theta[148] 3.091673e-01 0.0003604362 0.02127624 2.675858e-01
## theta[149] 2.506531e-01 0.0003224753 0.01759739 2.171541e-01
## theta[150] 1.402044e-01 0.0009037612 0.04541366 6.660664e-02
## theta[151] 2.453281e-01 0.0007908675 0.04134567 1.662480e-01
## theta[152] 1.437183e-01 0.0009192728 0.04870345 5.983902e-02
## theta[153] 1.513139e-01 0.0007136340 0.04087248 7.795456e-02
## theta[154] 1.398376e-01 0.0010072648 0.04984873 5.938809e-02
## theta[155] 1.426065e-01 0.0008732763 0.04654052 6.620470e-02
## theta[156] 2.330624e-01 0.0003417218 0.01886603 1.970631e-01
## theta[157] 2.260032e-01 0.0003937523 0.02111772 1.847318e-01
## theta[158] 2.343562e-01 0.0003308126 0.02163509 1.935117e-01
## theta[159] 1.243460e-01 0.0009230575 0.04347255 5.591978e-02
## theta[160] 2.428728e-01 0.0005095756 0.02869554 1.880748e-01
## theta[161] 1.248203e-01 0.0008987564 0.04644299 4.945994e-02
## theta[162] 1.589880e-01 0.0006565675 0.03678434 9.534669e-02
## theta[163] 1.098180e-01 0.0009794894 0.04247745 4.383325e-02
## theta[164] 2.582426e-01 0.0006030364 0.03234116 1.982757e-01
## theta[165] 2.705617e-01 0.0005299073 0.02846164 2.151561e-01
## theta[166] 2.292951e-01 0.0006940360 0.03632319 1.590217e-01
## theta[167] 2.813891e-01 0.0003173936 0.01722429 2.508542e-01
## theta[168] 2.538230e-01 0.0003890397 0.02131288 2.120198e-01
## theta[169] 2.802820e-01 0.0003559277 0.02025424 2.403289e-01
## theta[170] 2.070014e-01 0.0005299285 0.02542045 1.600386e-01
## theta[171] 1.187602e-01 0.0009463254 0.04359390 4.721679e-02
## theta[172] 1.105208e-01 0.0005532799 0.03220053 5.558625e-02
## theta[173] 1.964578e-01 0.0005654197 0.03011977 1.417266e-01
## theta[174] 2.675207e-01 0.0003566049 0.01834765 2.317706e-01
## theta[175] 2.840126e-01 0.0004197383 0.02038442 2.449638e-01
## theta[176] 2.555310e-01 0.0007414964 0.04056001 1.799897e-01
## theta[177] 2.650660e-01 0.0003951481 0.02490178 2.187682e-01
## theta[178] 2.511188e-01 0.0005968343 0.03285582 1.909228e-01
## theta[179] 2.618252e-01 0.0003540580 0.01988684 2.225930e-01
## theta[180] 2.049708e-01 0.0008492657 0.03908521 1.338929e-01
## theta[181] 2.399914e-01 0.0004308485 0.02375460 1.960566e-01
## theta[182] 2.768972e-01 0.0003544875 0.01941876 2.403597e-01
## theta[183] 1.270847e-01 0.0009843957 0.04589196 5.219367e-02
## theta[184] 1.197778e-01 0.0009358827 0.04417988 4.752487e-02
## theta[185] 2.917971e-01 0.0004089240 0.02136381 2.526456e-01
## theta[186] 2.371726e-01 0.0004524670 0.02504715 1.907284e-01
## theta[187] 1.223088e-01 0.0009053724 0.04393495 5.026616e-02
## theta[188] 1.335359e-01 0.0008309767 0.04189247 6.117886e-02
## theta[189] 1.200430e-01 0.0008858105 0.04678525 4.313703e-02
## theta[190] 1.373276e-01 0.0009003535 0.04514745 6.045376e-02
## theta[191] 1.944752e-01 0.0004169788 0.02195691 1.529231e-01
## theta[192] 2.377039e-01 0.0005515814 0.02859782 1.849887e-01
## theta[193] 1.456814e-01 0.0006833902 0.03963587 7.850414e-02
## theta[194] 1.130727e-01 0.0006856088 0.03658751 5.262050e-02
## theta[195] 1.221789e-01 0.0011043716 0.04817895 4.295084e-02
## theta[196] 1.242779e-01 0.0007515220 0.04635828 4.645629e-02
## theta[197] 1.713695e-01 0.0008526820 0.03690168 1.060449e-01
## theta[198] 1.233455e-01 0.0010101757 0.04510952 4.887749e-02
## theta[199] 1.061715e-01 0.0005427765 0.02672981 5.983347e-02
## theta[200] 2.347802e-01 0.0006939128 0.03659137 1.656970e-01
## theta[201] 2.208857e-01 0.0003389979 0.01859027 1.842127e-01
## theta[202] 2.315582e-01 0.0003627417 0.01863774 1.952597e-01
## theta[203] 2.592278e-01 0.0003195396 0.01901611 2.225721e-01
## theta[204] 2.270524e-01 0.0003819202 0.02232508 1.832718e-01
## theta[205] 2.649919e-01 0.0003038052 0.01677819 2.320800e-01
## theta[206] 1.586710e-01 0.0009295907 0.04910754 7.899814e-02
## theta[207] 2.271356e-01 0.0002728130 0.01714462 1.938272e-01
## theta[208] 1.786844e-01 0.0006308770 0.03446392 1.184028e-01
## theta[209] 2.590883e-01 0.0005554450 0.03301297 1.990179e-01
## theta[210] 1.173206e-01 0.0007499153 0.04439655 4.494610e-02
## theta[211] 2.336803e-01 0.0006967469 0.03877640 1.642284e-01
## theta[212] 2.405323e-01 0.0007237478 0.03700699 1.709183e-01
## theta[213] 2.377156e-01 0.0006187976 0.03526598 1.731241e-01
## theta[214] 2.679283e-01 0.0003110206 0.01803260 2.327445e-01
## theta[215] 1.258348e-01 0.0009194707 0.04595183 5.097868e-02
## theta[216] 1.042663e-01 0.0008070316 0.04040961 3.672797e-02
## theta[217] 1.197244e-01 0.0007702057 0.04416961 4.669089e-02
## theta[218] 2.904605e-01 0.0004624992 0.02483479 2.431526e-01
## theta[219] 1.172607e-01 0.0008879588 0.04274108 4.726301e-02
## theta[220] 1.194167e-01 0.0011118379 0.04760227 4.175532e-02
## theta[221] 2.348864e-01 0.0002810764 0.01554296 2.034648e-01
## theta[222] 1.253228e-01 0.0009081374 0.04580411 5.110447e-02
## theta[223] 1.192002e-01 0.0006700038 0.03849628 5.469861e-02
## theta[224] 1.181304e-01 0.0009005103 0.04417264 4.500488e-02
## theta[225] 1.248459e-01 0.0010211786 0.04423132 5.024285e-02
## theta[226] 2.698560e-01 0.0006996278 0.03994231 1.956394e-01
## theta[227] 1.266727e-01 0.0008765601 0.04608960 5.155386e-02
## theta[228] 2.257320e-01 0.0002658811 0.01713586 1.938075e-01
## theta[229] 1.252891e-01 0.0009321698 0.04645058 5.014046e-02
## theta[230] 2.318928e-01 0.0003287551 0.01890310 1.931952e-01
## theta[231] 1.352360e-01 0.0008165353 0.04308042 6.413290e-02
## theta[232] 2.844198e-01 0.0002983037 0.01733470 2.515089e-01
## theta[233] 2.448282e-01 0.0004178778 0.02126175 2.042416e-01
## theta[234] 2.480185e-01 0.0002320500 0.01602310 2.179989e-01
## theta[235] 2.126171e-01 0.0005382454 0.02644015 1.631439e-01
## theta[236] 2.344937e-01 0.0005468347 0.03089436 1.781098e-01
## theta[237] 2.271102e-01 0.0004865594 0.02795374 1.735679e-01
## theta[238] 2.535363e-01 0.0004687610 0.02650773 2.005228e-01
## theta[239] 2.727413e-01 0.0006912898 0.03923492 2.016760e-01
## theta[240] 2.296660e-01 0.0004953658 0.02999929 1.728126e-01
## theta[241] 2.526742e-01 0.0003005669 0.01642310 2.186458e-01
## theta[242] 2.650257e-01 0.0006464319 0.03252940 2.051846e-01
## theta[243] 2.797420e-01 0.0004902748 0.02813665 2.269198e-01
## theta[244] 2.622209e-01 0.0002985123 0.01765937 2.288871e-01
## theta[245] 2.869008e-01 0.0003014204 0.01794689 2.530808e-01
## theta[246] 1.507788e-01 0.0011086107 0.05391367 6.599536e-02
## theta[247] 1.251779e-01 0.0009265825 0.04727456 4.499076e-02
## theta[248] 2.486861e-01 0.0006530842 0.03586967 1.844640e-01
## theta[249] 2.939477e-01 0.0003095059 0.02011766 2.553424e-01
## theta[250] 3.014059e-01 0.0004051690 0.02126559 2.612688e-01
## theta[251] 1.199652e-01 0.0008484625 0.04380215 4.591508e-02
## theta[252] 2.502504e-01 0.0004617575 0.02700756 1.977171e-01
## theta[253] 2.992394e-01 0.0003488107 0.01821942 2.642078e-01
## theta[254] 2.552625e-01 0.0003144852 0.01725831 2.225313e-01
## theta[255] 2.304782e-01 0.0003897593 0.02000523 1.920602e-01
## theta[256] 2.643019e-01 0.0003088468 0.01714821 2.310939e-01
## theta[257] 2.791866e-01 0.0003433302 0.01819165 2.436849e-01
## theta[258] 2.619903e-01 0.0008544311 0.04561110 1.820870e-01
## theta[259] 1.227808e-01 0.0008197595 0.04686181 4.622399e-02
## theta[260] 3.078072e-01 0.0002917537 0.01743810 2.727859e-01
## theta[261] 1.570182e-01 0.0010332978 0.04945144 7.241171e-02
## theta[262] 2.138226e-01 0.0004683130 0.02421194 1.682721e-01
## theta[263] 2.457663e-01 0.0003062182 0.01766192 2.134914e-01
## theta[264] 2.215108e-01 0.0004404898 0.02666229 1.710936e-01
## theta[265] 2.551165e-01 0.0004266610 0.02389796 2.112572e-01
## theta[266] 2.914357e-01 0.0003496547 0.01955913 2.534953e-01
## theta[267] 2.565080e-01 0.0007475000 0.04421443 1.753840e-01
## theta[268] 1.358775e-01 0.0006961302 0.03693359 7.125460e-02
## theta[269] 2.627894e-01 0.0006887785 0.03402371 1.998403e-01
## theta[270] 2.818396e-01 0.0004132076 0.02468529 2.344220e-01
## theta[271] 2.439818e-01 0.0004614930 0.02573163 1.945520e-01
## theta[272] 2.204499e-01 0.0004966540 0.02851654 1.653827e-01
## theta[273] 2.333019e-01 0.0005038187 0.02911418 1.800351e-01
## theta[274] 2.834408e-01 0.0004269696 0.02238522 2.424216e-01
## theta[275] 1.214287e-01 0.0010438464 0.04676679 4.487328e-02
## theta[276] 2.139935e-01 0.0005378694 0.02909827 1.602317e-01
## theta[277] 1.238450e-01 0.0009242656 0.04614039 4.642173e-02
## theta[278] 1.633287e-01 0.0007498153 0.04131425 9.127198e-02
## theta[279] 1.246340e-01 0.0010103572 0.04732975 4.885316e-02
## theta[280] 2.196875e-01 0.0003604856 0.01706379 1.876962e-01
## theta[281] 2.499014e-01 0.0005434411 0.03038491 1.939430e-01
## theta[282] 1.247025e-01 0.0009855562 0.04714978 5.161484e-02
## theta[283] 2.295436e-01 0.0002412649 0.01639805 1.973840e-01
## theta[284] 1.261661e-01 0.0009288769 0.04631091 5.033488e-02
## theta[285] 2.827849e-01 0.0003309369 0.01745551 2.511439e-01
## theta[286] 9.927413e-02 0.0005162321 0.02841240 4.950013e-02
## theta[287] 2.298240e-01 0.0004675371 0.02853346 1.765916e-01
## theta[288] 2.698940e-01 0.0005918789 0.03456205 2.055539e-01
## theta[289] 1.255836e-01 0.0010015135 0.04914739 4.701508e-02
## theta[290] 2.758194e-01 0.0003079182 0.01743101 2.432376e-01
## theta[291] 1.817829e-01 0.0007416320 0.03636417 1.172307e-01
## theta[292] 2.466245e-01 0.0006560835 0.03455742 1.809213e-01
## theta[293] 2.217299e-01 0.0007454469 0.04165257 1.446620e-01
## theta[294] 2.508956e-01 0.0003962048 0.02637238 2.007555e-01
## theta[295] 2.433593e-01 0.0004319513 0.02266182 1.997553e-01
## theta[296] 1.266627e-01 0.0009595600 0.04988349 4.327263e-02
## theta[297] 2.810924e-01 0.0002611062 0.01689030 2.483816e-01
## theta[298] 2.589814e-01 0.0004500056 0.02391868 2.144702e-01
## theta[299] 2.278117e-01 0.0005048477 0.02707419 1.763415e-01
## theta[300] 2.186543e-01 0.0004126807 0.02572151 1.692782e-01
## theta[301] 2.745899e-01 0.0004110090 0.02375741 2.284566e-01
## theta[302] 2.401685e-01 0.0007132554 0.03925807 1.704889e-01
## theta[303] 2.481625e-01 0.0007762338 0.04110776 1.767359e-01
## theta[304] 1.240859e-01 0.0009429230 0.04540540 5.222677e-02
## theta[305] 2.227247e-01 0.0004691475 0.02667906 1.744497e-01
## theta[306] 2.348776e-01 0.0002931553 0.01720681 2.018534e-01
## theta[307] 2.041022e-01 0.0005403026 0.02978449 1.455429e-01
## theta[308] 1.733159e-01 0.0009335563 0.04408421 9.265034e-02
## theta[309] 2.379118e-01 0.0004233822 0.02163595 1.958416e-01
## theta[310] 2.842800e-01 0.0003303394 0.01875593 2.477984e-01
## theta[311] 1.196638e-01 0.0006765253 0.03504961 6.150112e-02
## theta[312] 2.415791e-01 0.0006176948 0.03453275 1.738164e-01
## theta[313] 2.449029e-01 0.0004931239 0.02896009 1.937907e-01
## theta[314] 1.145377e-01 0.0010063702 0.04641686 4.085831e-02
## theta[315] 1.382499e-01 0.0009580595 0.04836192 5.677439e-02
## theta[316] 2.539780e-01 0.0008302252 0.04350758 1.748494e-01
## theta[317] 2.798447e-01 0.0004244396 0.02248292 2.372443e-01
## theta[318] 2.467980e-01 0.0003586294 0.01948843 2.099401e-01
## theta[319] 2.598570e-01 0.0005059441 0.02840305 2.064369e-01
## theta[320] 1.287369e-01 0.0009034590 0.04506012 5.381265e-02
## theta[321] 1.404266e-01 0.0010667653 0.04797057 6.308936e-02
## theta[322] 1.324751e-01 0.0007077556 0.04284861 5.839273e-02
## theta[323] 2.427203e-01 0.0004166503 0.02740221 1.915561e-01
## theta[324] 1.234092e-01 0.0009153951 0.04502222 4.797047e-02
## theta[325] 1.266362e-01 0.0009704170 0.04845100 4.999043e-02
## theta[326] 2.341894e-01 0.0007745282 0.04458907 1.538949e-01
## theta[327] 1.222969e-01 0.0009958682 0.04392552 4.990679e-02
## theta[328] 2.388627e-01 0.0007559217 0.03756572 1.723997e-01
## theta[329] 2.579679e-01 0.0005593055 0.02796656 2.049606e-01
## theta[330] 1.443759e-01 0.0011165937 0.05245755 6.104062e-02
## theta[331] 2.573520e-01 0.0003273359 0.01745793 2.239260e-01
## theta[332] 2.354001e-01 0.0004565298 0.02420425 1.865667e-01
## theta[333] 2.277835e-01 0.0003660259 0.01865922 1.922328e-01
## theta[334] 2.697017e-01 0.0003458245 0.01873405 2.336055e-01
## theta[335] 1.250755e-01 0.0009829960 0.04773730 4.890407e-02
## theta[336] 1.285390e-01 0.0009596759 0.04674786 4.889400e-02
## theta[337] 2.279272e-01 0.0007661319 0.04025254 1.531161e-01
## theta[338] 1.260020e-01 0.0008981114 0.04544729 5.369774e-02
## theta[339] 2.198742e-01 0.0004712329 0.02597661 1.705586e-01
## theta[340] 2.068389e-01 0.0003611864 0.02058318 1.667511e-01
## theta[341] 2.676086e-01 0.0005541234 0.02806859 2.136649e-01
## theta[342] 2.699645e-01 0.0003623265 0.01980259 2.312263e-01
## theta[343] 2.839143e-01 0.0003475384 0.01811219 2.474235e-01
## theta[344] 2.432189e-01 0.0006224752 0.03711334 1.789561e-01
## theta[345] 1.089520e-01 0.0005620279 0.03025521 5.635040e-02
## theta[346] 2.363380e-01 0.0002850915 0.01657132 2.037655e-01
## theta[347] 2.153296e-01 0.0004544851 0.02644172 1.664174e-01
## theta[348] 1.248544e-01 0.0008379013 0.04557860 5.045911e-02
## theta[349] 2.781761e-01 0.0003699928 0.02170925 2.367592e-01
## theta[350] 2.461304e-01 0.0003467372 0.01865189 2.114771e-01
## theta[351] 1.382585e-01 0.0009035813 0.04325670 6.790873e-02
## theta[352] 2.327173e-01 0.0009227873 0.04555405 1.479926e-01
## theta[353] 2.253333e-01 0.0007585426 0.04086832 1.520643e-01
## theta[354] 2.571127e-01 0.0002622945 0.01588394 2.268108e-01
## theta[355] 2.678999e-01 0.0007826799 0.04673376 1.840129e-01
## theta[356] 2.386168e-01 0.0005603351 0.03301894 1.767879e-01
## theta[357] 1.557871e-01 0.0009876128 0.04896338 7.247424e-02
## theta[358] 2.684663e-01 0.0004424296 0.02623357 2.190497e-01
## theta[359] 1.255967e-01 0.0008680934 0.04554556 4.946419e-02
## theta[360] 2.185041e-01 0.0007682006 0.03430059 1.563338e-01
## theta[361] 2.568937e-01 0.0007931820 0.04358501 1.761067e-01
## theta[362] 1.998242e-01 0.0005946412 0.02944612 1.425113e-01
## theta[363] 1.233266e-01 0.0009286840 0.04541189 4.833835e-02
## theta[364] 1.207411e-01 0.0004656623 0.02987358 7.019420e-02
## theta[365] 2.822082e-01 0.0003184214 0.01663437 2.503288e-01
## theta[366] 2.355446e-01 0.0004482962 0.02620565 1.877120e-01
## theta[367] 2.537332e-01 0.0002568215 0.01665360 2.240473e-01
## theta[368] 2.859208e-01 0.0003622157 0.01774167 2.515025e-01
## theta[369] 1.031267e-01 0.0006260128 0.02973593 5.209949e-02
## theta[370] 2.550665e-01 0.0002991300 0.01567456 2.254057e-01
## theta[371] 2.216270e-01 0.0004457575 0.02529326 1.728322e-01
## theta[372] 2.784362e-01 0.0002705868 0.01589873 2.467090e-01
## theta[373] 2.912594e-01 0.0006810911 0.03394536 2.272708e-01
## theta[374] 2.418983e-01 0.0006481674 0.03523349 1.777079e-01
## theta[375] 1.338950e-01 0.0009354067 0.04700611 5.566788e-02
## theta[376] 2.464377e-01 0.0004181204 0.02198339 2.052893e-01
## theta[377] 9.974717e-02 0.0009181533 0.03598834 4.250031e-02
## theta[378] 2.676606e-01 0.0004314546 0.02168951 2.265852e-01
## theta[379] 1.823818e-01 0.0007708331 0.04202096 1.097933e-01
## theta[380] 1.183319e-01 0.0007235691 0.03871473 5.391733e-02
## theta[381] 1.219595e-01 0.0009113524 0.04626146 4.544578e-02
## theta[382] 2.226147e-01 0.0005500459 0.03450713 1.556028e-01
## theta[383] 1.168255e-01 0.0008918224 0.04112240 4.550397e-02
## theta[384] 1.248012e-01 0.0005830838 0.03221642 6.414536e-02
## theta[385] 2.458646e-01 0.0007460498 0.04162581 1.675646e-01
## theta[386] 2.494624e-01 0.0003436325 0.01796441 2.156762e-01
## theta[387] 1.360747e-01 0.0006225174 0.03349868 7.557169e-02
## theta[388] 1.247262e-01 0.0010608925 0.04739957 4.633410e-02
## theta[389] 1.345225e-01 0.0008885644 0.04570979 5.975622e-02
## theta[390] 2.645645e-01 0.0003312630 0.01805451 2.306026e-01
## theta[391] 1.263548e-01 0.0009467373 0.04564552 4.742712e-02
## theta[392] 1.321724e-01 0.0008394261 0.04503411 5.632948e-02
## theta[393] 1.270719e-01 0.0008779139 0.04713243 5.240843e-02
## theta[394] 2.587569e-01 0.0004103246 0.02205863 2.164561e-01
## theta[395] 1.140744e-01 0.0009199041 0.04281245 4.465245e-02
## theta[396] 2.250105e-01 0.0002779015 0.01828855 1.903167e-01
## theta[397] 1.905185e-01 0.0006361942 0.03105598 1.338206e-01
## theta[398] 1.937298e-01 0.0004449140 0.02236294 1.521736e-01
## theta[399] 1.178630e-01 0.0008115373 0.04383584 4.586691e-02
## theta[400] 2.595042e-01 0.0006014709 0.03419502 1.947675e-01
## theta[401] 2.517010e-01 0.0004158907 0.02355373 2.070057e-01
## theta[402] 2.239396e-01 0.0005113124 0.02844300 1.717632e-01
## theta[403] 2.415528e-01 0.0005494468 0.03166189 1.832089e-01
## theta[404] 1.184106e-01 0.0009022199 0.04440650 4.683302e-02
## theta[405] 2.680779e-01 0.0002631799 0.01706093 2.352319e-01
## theta[406] 2.560949e-01 0.0003001073 0.01631094 2.233061e-01
## theta[407] 2.528676e-01 0.0003017720 0.01752173 2.192209e-01
## theta[408] 1.883096e-01 0.0009357176 0.04499654 1.094004e-01
## theta[409] 2.617407e-01 0.0008980086 0.04580852 1.757113e-01
## theta[410] 2.412849e-01 0.0007587469 0.04331686 1.594528e-01
## theta[411] 1.553975e-01 0.0008281243 0.04745311 7.685334e-02
## theta[412] 2.525327e-01 0.0003209483 0.01724449 2.194135e-01
## theta[413] 2.469796e-01 0.0004212665 0.02549441 1.996036e-01
## theta[414] 2.892914e-01 0.0003951303 0.02201246 2.468661e-01
## theta[415] 2.554343e-01 0.0005690596 0.02741057 2.048032e-01
## theta[416] 1.376039e-01 0.0009160603 0.04653479 6.187104e-02
## theta[417] 2.433929e-01 0.0003671882 0.02079138 2.044352e-01
## theta[418] 2.471982e-01 0.0004356486 0.02500728 1.989260e-01
## theta[419] 2.204722e-01 0.0006500876 0.03516729 1.565412e-01
## theta[420] 2.329401e-01 0.0007727643 0.03667579 1.616657e-01
## theta[421] 1.036094e-01 0.0007377794 0.03189497 4.804407e-02
## theta[422] 2.387806e-01 0.0003011245 0.01615207 2.082594e-01
## theta[423] 1.193846e-01 0.0006727873 0.03503619 6.056641e-02
## theta[424] 3.102135e-01 0.0003791060 0.02137971 2.708419e-01
## theta[425] 1.220117e-01 0.0008082304 0.04319428 5.019190e-02
## theta[426] 2.237711e-01 0.0004526541 0.02429728 1.782742e-01
## theta[427] 1.147027e-01 0.0006310833 0.03208688 6.124126e-02
## theta[428] 1.260501e-01 0.0008113381 0.04383472 5.411880e-02
## theta[429] 2.254522e-01 0.0003202409 0.01808563 1.898694e-01
## theta[430] 1.210779e-01 0.0008851763 0.04334107 4.980226e-02
## theta[431] 1.410309e-01 0.0009778541 0.04797349 5.840187e-02
## theta[432] 1.291148e-01 0.0008140578 0.04652270 5.042147e-02
## theta[433] 1.121665e-01 0.0006156415 0.03532287 5.050650e-02
## theta[434] 1.198934e-01 0.0006958862 0.03714707 5.659531e-02
## theta[435] 1.235800e-01 0.0009339498 0.04909907 4.530062e-02
## theta[436] 2.567579e-01 0.0007105411 0.03861163 1.854609e-01
## theta[437] 1.411455e-01 0.0008970834 0.04764433 6.082147e-02
## theta[438] 2.359218e-01 0.0006890594 0.03683774 1.668794e-01
## theta[439] 1.204064e-01 0.0009830331 0.04534640 4.435440e-02
## theta[440] 2.189361e-01 0.0002877819 0.02226940 1.761165e-01
## theta[441] 2.472272e-01 0.0004271751 0.02172871 2.064581e-01
## theta[442] 2.574138e-01 0.0003296691 0.01735586 2.244114e-01
## theta[443] 1.272482e-01 0.0009422534 0.04871499 4.703768e-02
## theta[444] 2.602063e-01 0.0007861047 0.04048751 1.868623e-01
## theta[445] 2.408686e-01 0.0002775219 0.01705772 2.081562e-01
## theta[446] 1.831480e-01 0.0009572733 0.04539716 1.051787e-01
## theta[447] 2.527991e-01 0.0005126692 0.02490407 2.062064e-01
## theta[448] 2.290012e-01 0.0007209415 0.03057508 1.731655e-01
## theta[449] 2.497885e-01 0.0003283295 0.01658866 2.177753e-01
## theta[450] 2.647125e-01 0.0003315384 0.02126835 2.230208e-01
## theta[451] 1.004319e-01 0.0006511837 0.02897296 5.111485e-02
## theta[452] 2.459073e-01 0.0008348054 0.04194996 1.689386e-01
## theta[453] 2.230847e-01 0.0004408064 0.02387587 1.786297e-01
## theta[454] 3.074992e-01 0.0003432843 0.01832707 2.724204e-01
## theta[455] 1.106499e-01 0.0006371118 0.03388548 5.175480e-02
## theta[456] 1.265835e-01 0.0008929140 0.04924971 4.770112e-02
## theta[457] 1.465783e-01 0.0009749223 0.04713167 6.879101e-02
## theta[458] 1.308527e-01 0.0009756855 0.04836253 4.905014e-02
## theta[459] 3.184585e-01 0.0003877008 0.02185692 2.767566e-01
## theta[460] 1.119131e-01 0.0007064125 0.03404607 5.249019e-02
## theta[461] 1.249763e-01 0.0008522418 0.04811976 5.099822e-02
## theta[462] 2.384151e-01 0.0003947566 0.02555297 1.917102e-01
## theta[463] 2.024421e-01 0.0003396259 0.01945569 1.643185e-01
## theta[464] 1.227516e-01 0.0007743532 0.04405077 5.139519e-02
## theta[465] 2.254337e-01 0.0006325412 0.03105265 1.675795e-01
## theta[466] 2.665928e-01 0.0004389341 0.02354264 2.214672e-01
## theta[467] 1.278112e-01 0.0008689337 0.04444792 5.345299e-02
## theta[468] 1.076514e-01 0.0005751568 0.02952582 5.422916e-02
## theta[469] 2.307458e-01 0.0004373202 0.02311751 1.866427e-01
## theta[470] 2.414714e-01 0.0004016566 0.02577001 1.934568e-01
## theta[471] 2.965119e-01 0.0003376841 0.01987964 2.607411e-01
## theta[472] 1.186099e-01 0.0010664813 0.04511504 4.468900e-02
## theta[473] 2.584397e-01 0.0004322156 0.02463135 2.138717e-01
## theta[474] 3.008530e-01 0.0004224148 0.02259871 2.583463e-01
## theta[475] 1.170584e-01 0.0008711018 0.04252047 4.336900e-02
## theta[476] 1.767797e-01 0.0006980413 0.03742691 1.098049e-01
## theta[477] 2.583985e-01 0.0003637711 0.02176729 2.170945e-01
## theta[478] 2.381593e-01 0.0005354781 0.03274744 1.772098e-01
## theta[479] 1.413426e-01 0.0006562663 0.03618051 7.777074e-02
## theta[480] 3.004520e-01 0.0003062249 0.01863175 2.649225e-01
## theta[481] 2.215018e-01 0.0003581163 0.02032016 1.837948e-01
## theta[482] 1.635119e-01 0.0007744474 0.03634369 1.014317e-01
## theta[483] 2.424541e-01 0.0004367378 0.02456922 1.947698e-01
## theta[484] 2.319821e-01 0.0006469578 0.03624178 1.637493e-01
## theta[485] 1.541854e-01 0.0009653538 0.04916498 7.419799e-02
## theta[486] 2.835347e-01 0.0003007843 0.01739240 2.479162e-01
## theta[487] 2.482064e-01 0.0005954512 0.03099821 1.891722e-01
## theta[488] 1.890031e-01 0.0006910259 0.03251857 1.287736e-01
## theta[489] 2.265206e-01 0.0005375730 0.02668959 1.761319e-01
## theta[490] 2.477438e-01 0.0003810090 0.02362664 2.018176e-01
## theta[491] 1.348479e-01 0.0010020683 0.04891985 5.264529e-02
## theta[492] 2.288978e-01 0.0003842958 0.02226675 1.870650e-01
## theta[493] 1.228551e-01 0.0008804696 0.04633239 4.852062e-02
## theta[494] 2.474696e-01 0.0004330751 0.01986340 2.086546e-01
## theta[495] 1.056399e-01 0.0006158455 0.03591454 4.535004e-02
## theta[496] 1.145804e-01 0.0007496596 0.03907192 4.976570e-02
## theta[497] 2.444321e-01 0.0003808167 0.02100442 2.056562e-01
## theta[498] 2.818430e-01 0.0003593358 0.01796325 2.471314e-01
## theta[499] 2.378327e-01 0.0004331832 0.02239089 1.956670e-01
## theta[500] 1.121223e-01 0.0005706260 0.02993516 5.909320e-02
## theta[501] 1.189738e-01 0.0008420557 0.04048388 4.764058e-02
## theta[502] 2.440518e-01 0.0002855559 0.01541543 2.156263e-01
## theta[503] 1.204365e-01 0.0008179775 0.04032757 5.315115e-02
## theta[504] 2.672883e-01 0.0004087572 0.02358298 2.224386e-01
## theta[505] 1.267408e-01 0.0008467618 0.04728727 4.890502e-02
## theta[506] 2.340230e-01 0.0002944438 0.02074539 1.954858e-01
## theta[507] 1.516759e-01 0.0009246768 0.04830981 7.175496e-02
## theta[508] 1.846999e-01 0.0005972437 0.02950896 1.285923e-01
## theta[509] 2.587081e-01 0.0003190922 0.01829759 2.224636e-01
## theta[510] 2.434386e-01 0.0004625830 0.02464706 1.969185e-01
## theta[511] 1.273889e-01 0.0006856193 0.03840765 5.907234e-02
## theta[512] 2.944768e-01 0.0003588227 0.01996877 2.563702e-01
## theta[513] 2.460267e-01 0.0003976635 0.02164054 2.054687e-01
## theta[514] 2.174400e-01 0.0004740562 0.02731498 1.657148e-01
## theta[515] 1.257572e-01 0.0009301517 0.04581801 5.139349e-02
## theta[516] 2.378055e-01 0.0006994294 0.04042774 1.614378e-01
## theta[517] 2.037175e-01 0.0005752979 0.03232739 1.423269e-01
## theta[518] 1.096558e-01 0.0009349773 0.04060790 4.387029e-02
## theta[519] 1.253803e-01 0.0009059628 0.04819198 4.760811e-02
## theta[520] 1.274488e-01 0.0009266755 0.04636733 5.012447e-02
## theta[521] 2.350005e-01 0.0003996713 0.01977588 1.949237e-01
## theta[522] 2.642708e-01 0.0003500699 0.01780526 2.323731e-01
## theta[523] 2.994574e-01 0.0003998666 0.02328303 2.540407e-01
## theta[524] 2.310089e-01 0.0006655164 0.03766105 1.621114e-01
## theta[525] 2.232671e-01 0.0003200905 0.01835254 1.882423e-01
## theta[526] 2.613293e-01 0.0004602767 0.02798525 2.084491e-01
## theta[527] 2.760032e-01 0.0003085863 0.01738794 2.421621e-01
## theta[528] 1.207849e-01 0.0007869427 0.04673575 4.501935e-02
## theta[529] 2.286816e-01 0.0002923418 0.01642870 1.979544e-01
## theta[530] 2.345685e-01 0.0004477118 0.02574681 1.836624e-01
## theta[531] 2.685118e-01 0.0003365193 0.01821386 2.333711e-01
## theta[532] 1.247516e-01 0.0008295813 0.04822150 5.032352e-02
## theta[533] 1.170306e-01 0.0006564784 0.03246193 6.000017e-02
## theta[534] 3.100857e-01 0.0005948174 0.02716185 2.590490e-01
## theta[535] 2.351559e-01 0.0008393497 0.04312166 1.587877e-01
## theta[536] 1.186034e-01 0.0008163393 0.04175010 4.811267e-02
## theta[537] 2.392573e-01 0.0003675636 0.01898762 2.034116e-01
## theta[538] 1.197852e-01 0.0008649094 0.04219237 4.866396e-02
## theta[539] 2.534016e-01 0.0004515729 0.02241525 2.135826e-01
## theta[540] 2.467197e-01 0.0007777870 0.03805857 1.767780e-01
## theta[541] 2.247630e-01 0.0008959983 0.04313948 1.463384e-01
## theta[542] 2.258011e-01 0.0006343575 0.03584838 1.576803e-01
## theta[543] 2.189790e-01 0.0007669277 0.03894016 1.448641e-01
## theta[544] 1.245616e-01 0.0007335303 0.03662689 5.849680e-02
## theta[545] 2.554057e-01 0.0007824190 0.04378423 1.742855e-01
## theta[546] 2.356166e-01 0.0003443043 0.01910495 1.993009e-01
## theta[547] 2.552041e-01 0.0007994559 0.04208644 1.775513e-01
## theta[548] 1.235558e-01 0.0008748113 0.04644189 4.670468e-02
## theta[549] 1.332946e-01 0.0007212851 0.03548209 7.228166e-02
## theta[550] 1.089246e-01 0.0005904104 0.02879665 5.941070e-02
## theta[551] 1.274780e-01 0.0007644237 0.04835473 4.948777e-02
## theta[552] 1.169880e-01 0.0008309902 0.04468580 4.423979e-02
## theta[553] 2.591763e-01 0.0002785800 0.01677360 2.265583e-01
## theta[554] 1.143793e-01 0.0008749429 0.04069963 4.594770e-02
## theta[555] 2.552884e-01 0.0005245389 0.03092016 2.004932e-01
## theta[556] 9.835805e-02 0.0006314589 0.03073413 4.671907e-02
## theta[557] 2.402216e-01 0.0008183738 0.04041463 1.643410e-01
## theta[558] 2.488116e-01 0.0005407800 0.02889180 1.932597e-01
## theta[559] 2.275014e-01 0.0007257934 0.03507154 1.636964e-01
## theta[560] 2.226537e-01 0.0005717592 0.03114373 1.637886e-01
## theta[561] 1.240193e-01 0.0008057209 0.04617503 4.753475e-02
## theta[562] 1.231067e-01 0.0009383703 0.04588831 4.700106e-02
## theta[563] 2.667788e-01 0.0004270485 0.02129058 2.277420e-01
## theta[564] 2.360895e-01 0.0003736796 0.02020643 1.967249e-01
## theta[565] 2.671526e-01 0.0007636948 0.04058092 1.909085e-01
## theta[566] 2.600569e-01 0.0005026358 0.02810744 2.071895e-01
## theta[567] 2.635467e-01 0.0004188668 0.02522843 2.145122e-01
## theta[568] 2.289458e-01 0.0004547970 0.02322319 1.848130e-01
## theta[569] 2.450683e-01 0.0003509919 0.01944590 2.083299e-01
## theta[570] 1.414060e-01 0.0005909196 0.03310200 8.340716e-02
## theta[571] 1.412210e-01 0.0008620738 0.04674793 6.111602e-02
## theta[572] 2.864230e-01 0.0004071965 0.02288139 2.428171e-01
## theta[573] 2.050757e-01 0.0007786885 0.03923469 1.330872e-01
## theta[574] 2.494781e-01 0.0008925232 0.04694058 1.663413e-01
## theta[575] 1.263078e-01 0.0009695854 0.04816531 4.906205e-02
## theta[576] 1.270221e-01 0.0008242363 0.04547314 5.242955e-02
## theta[577] 2.293027e-01 0.0004072170 0.02263053 1.856806e-01
## theta[578] 2.330509e-01 0.0008171726 0.04234954 1.546434e-01
## theta[579] 1.262553e-01 0.0008501492 0.04675420 4.962363e-02
## theta[580] 1.352104e-01 0.0009897033 0.04703947 5.914333e-02
## theta[581] 2.316382e-01 0.0007369464 0.03912274 1.596150e-01
## theta[582] 2.356778e-01 0.0004021784 0.02102785 1.961011e-01
## theta[583] 2.563608e-01 0.0004385464 0.02989743 2.009590e-01
## theta[584] 1.494221e-01 0.0005468330 0.03348754 9.344021e-02
## theta[585] 2.557950e-01 0.0003916982 0.02145602 2.159044e-01
## theta[586] 1.310376e-01 0.0007814110 0.04485624 5.491511e-02
## theta[587] 2.577353e-01 0.0004645436 0.02715352 2.076383e-01
## theta[588] 1.274918e-01 0.0009153070 0.04758117 4.903751e-02
## theta[589] 2.322425e-01 0.0009086939 0.04603685 1.449936e-01
## theta[590] 1.164250e-01 0.0007146821 0.03397900 5.693103e-02
## theta[591] 2.979716e-01 0.0002882952 0.01691050 2.653193e-01
## theta[592] 8.234886e-02 0.0006512714 0.02614292 3.916180e-02
## theta[593] 2.244830e-01 0.0005519939 0.03038611 1.692834e-01
## theta[594] 2.568644e-01 0.0003594766 0.01916813 2.207957e-01
## theta[595] 2.560018e-01 0.0004892631 0.02758834 2.048471e-01
## theta[596] 2.271742e-01 0.0003395111 0.01818026 1.926735e-01
## theta[597] 1.488881e-01 0.0007909579 0.04383936 7.506547e-02
## theta[598] 2.509577e-01 0.0003231963 0.01765559 2.178467e-01
## theta[599] 2.651500e-01 0.0003069987 0.01701613 2.328976e-01
## theta[600] 2.314386e-01 0.0005729838 0.02775427 1.780174e-01
## theta[601] 2.593078e-01 0.0004192226 0.02493075 2.120526e-01
## theta[602] 2.938092e-01 0.0003154757 0.01704518 2.610068e-01
## theta[603] 2.381452e-01 0.0005014678 0.02468414 1.915700e-01
## theta[604] 2.506116e-01 0.0006170606 0.03516560 1.876220e-01
## theta[605] 1.614243e-01 0.0006253683 0.03684078 9.942763e-02
## theta[606] 2.525863e-01 0.0006436380 0.03120370 1.935408e-01
## theta[607] 1.224579e-01 0.0009532630 0.04704794 4.680234e-02
## theta[608] 1.568966e-01 0.0007194232 0.03368355 9.617261e-02
## theta[609] 2.821227e-01 0.0004234063 0.02338143 2.354152e-01
## theta[610] 2.395350e-01 0.0005112314 0.03248179 1.781419e-01
## theta[611] 2.359015e-01 0.0005892143 0.03014064 1.796396e-01
## theta[612] 2.712806e-01 0.0005028000 0.03069093 2.159847e-01
## theta[613] 2.189210e-01 0.0004281874 0.02558186 1.702828e-01
## theta[614] 1.058254e-01 0.0008036380 0.03359284 4.805018e-02
## theta[615] 2.417485e-01 0.0005695106 0.03203937 1.808775e-01
## theta[616] 1.183477e-01 0.0008947414 0.04469580 4.246355e-02
## theta[617] 1.832960e-01 0.0011279886 0.04824570 1.014183e-01
## theta[618] 2.889185e-01 0.0003302226 0.01710933 2.565239e-01
## theta[619] 2.939509e-01 0.0003208851 0.02057443 2.552334e-01
## theta[620] 2.479454e-01 0.0006569982 0.03353239 1.869342e-01
## theta[621] 2.362167e-01 0.0005727127 0.03118165 1.807910e-01
## theta[622] 1.199868e-01 0.0008388382 0.04553843 4.563411e-02
## theta[623] 2.017309e-01 0.0005458897 0.02908970 1.462310e-01
## theta[624] 2.475720e-01 0.0003302772 0.01680742 2.159848e-01
## theta[625] 2.441535e-01 0.0003292920 0.01885058 2.090306e-01
## theta[626] 2.633316e-01 0.0005191776 0.02967666 2.084827e-01
## theta[627] 1.233649e-01 0.0009275205 0.04596195 4.905464e-02
## theta[628] 3.276673e-01 0.0003313037 0.01898308 2.922593e-01
## theta[629] 2.592714e-01 0.0008429265 0.04831486 1.693359e-01
## theta[630] 1.229975e-01 0.0009702247 0.04629130 5.016776e-02
## theta[631] 2.718211e-01 0.0003073283 0.01720379 2.388642e-01
## theta[632] 1.249951e-01 0.0008264102 0.04663202 4.829556e-02
## theta[633] 2.833216e-01 0.0002879255 0.01790069 2.486217e-01
## theta[634] 2.601555e-01 0.0003402312 0.02060969 2.211122e-01
## theta[635] 2.089687e-01 0.0005264938 0.02633085 1.604393e-01
## theta[636] 2.352242e-01 0.0004441801 0.02353770 1.928818e-01
## theta[637] 2.485170e-01 0.0002793805 0.01791970 2.136969e-01
## theta[638] 2.729242e-01 0.0003546094 0.01703499 2.384929e-01
## theta[639] 1.152915e-01 0.0008796178 0.04271118 4.216991e-02
## theta[640] 3.199580e-01 0.0003581196 0.01741096 2.868799e-01
## theta[641] 2.142750e-01 0.0004697059 0.02671150 1.616139e-01
## theta[642] 2.780881e-01 0.0003512905 0.01876225 2.411017e-01
## theta[643] 2.259063e-01 0.0004038352 0.02027317 1.866738e-01
## theta[644] 2.500581e-01 0.0002999889 0.01740278 2.169956e-01
## theta[645] 2.612560e-01 0.0004320622 0.02649553 2.095265e-01
## theta[646] 2.280478e-01 0.0004427262 0.02546365 1.786784e-01
## theta[647] 2.205863e-01 0.0007176156 0.03540631 1.527138e-01
## theta[648] 1.246899e-01 0.0007937254 0.04418094 5.249446e-02
## theta[649] 1.143927e-01 0.0006996287 0.03618073 5.370084e-02
## theta[650] 2.660037e-01 0.0005064933 0.03284892 2.047984e-01
## theta[651] 2.182638e-01 0.0009805226 0.03998105 1.450922e-01
## theta[652] 1.243029e-01 0.0007901452 0.04684321 4.788651e-02
## theta[653] 2.231588e-01 0.0006384165 0.03517496 1.585847e-01
## theta[654] 9.861983e-02 0.0005481556 0.02933490 5.047095e-02
## theta[655] 2.822352e-01 0.0003431393 0.01931043 2.455981e-01
## theta[656] 2.437599e-01 0.0002939676 0.01662849 2.117145e-01
## theta[657] 2.300840e-01 0.0003422604 0.02010575 1.926257e-01
## theta[658] 2.027990e-01 0.0004998424 0.02934410 1.487544e-01
## theta[659] 2.262897e-01 0.0006272313 0.03608000 1.553969e-01
## theta[660] 1.343186e-01 0.0007823707 0.04680525 5.633398e-02
## theta[661] 3.168970e-01 0.0003469713 0.01924327 2.809856e-01
## theta[662] 1.213364e-01 0.0008439389 0.04325452 4.760936e-02
## theta[663] 2.229814e-01 0.0006171405 0.03417361 1.562512e-01
## theta[664] 2.706762e-01 0.0003758009 0.02160730 2.282692e-01
## theta[665] 2.398473e-01 0.0004533367 0.02702985 1.882268e-01
## theta[666] 2.158707e-01 0.0005580157 0.02900448 1.623781e-01
## theta[667] 2.422733e-01 0.0004135995 0.02101237 2.051745e-01
## theta[668] 2.573840e-01 0.0003960836 0.02381423 2.140639e-01
## theta[669] 2.349397e-01 0.0007403378 0.03950537 1.647882e-01
## theta[670] 1.278543e-01 0.0009022732 0.04758914 4.829114e-02
## theta[671] 1.154053e-01 0.0006536542 0.03645609 5.434761e-02
## theta[672] 1.252608e-01 0.0009199166 0.04767749 4.840640e-02
## theta[673] 2.740584e-01 0.0003093350 0.01839765 2.393979e-01
## theta[674] 1.252137e-01 0.0009675334 0.04618979 4.801660e-02
## theta[675] 2.375789e-01 0.0007230515 0.03846461 1.656885e-01
## theta[676] 1.810122e-01 0.0003876442 0.02234876 1.390593e-01
## theta[677] 2.330866e-01 0.0004880322 0.03089974 1.737314e-01
## theta[678] 2.901019e-01 0.0003422608 0.01998182 2.528595e-01
## theta[679] 2.328267e-01 0.0005319666 0.02720689 1.823397e-01
## theta[680] 2.689726e-01 0.0003090477 0.01671041 2.356555e-01
## theta[681] 1.231731e-01 0.0008946619 0.04605916 4.724765e-02
## theta[682] 2.719920e-01 0.0006242016 0.03429759 2.062871e-01
## theta[683] 2.431128e-01 0.0004257315 0.02168041 2.022088e-01
## theta[684] 2.706637e-01 0.0002829432 0.01637475 2.378060e-01
## theta[685] 2.456035e-01 0.0004746266 0.02571211 1.956066e-01
## theta[686] 2.404567e-01 0.0004639454 0.02761189 1.894589e-01
## theta[687] 2.289642e-01 0.0008469122 0.04457272 1.486059e-01
## theta[688] 2.163660e-01 0.0003980970 0.02141730 1.745251e-01
## theta[689] 2.790948e-01 0.0003675243 0.02043755 2.407695e-01
## theta[690] 1.210152e-01 0.0006753539 0.03473798 6.037996e-02
## theta[691] 2.796657e-01 0.0002962201 0.01791137 2.447060e-01
## theta[692] 2.068677e-01 0.0004841365 0.02657178 1.565833e-01
## theta[693] 2.903832e-01 0.0003006319 0.01841347 2.560487e-01
## theta[694] 9.469594e-02 0.0006182105 0.02746032 4.617733e-02
## theta[695] 2.465286e-01 0.0003289729 0.01693782 2.142637e-01
## theta[696] 2.817351e-01 0.0004681302 0.02313159 2.369245e-01
## theta[697] 2.289953e-01 0.0004920232 0.02755995 1.773856e-01
## theta[698] 2.847241e-01 0.0006601191 0.03953908 2.124536e-01
## theta[699] 2.709862e-01 0.0006198286 0.03516872 2.047543e-01
## theta[700] 2.322830e-01 0.0003868540 0.02542250 1.848533e-01
## theta[701] 1.251577e-01 0.0008716910 0.04535691 5.323314e-02
## theta[702] 1.347414e-01 0.0009692311 0.04725060 5.755166e-02
## theta[703] 2.562391e-01 0.0003864298 0.02221564 2.144338e-01
## theta[704] 3.064549e-01 0.0003128789 0.01765827 2.716440e-01
## theta[705] 2.543231e-01 0.0003757126 0.02208537 2.125290e-01
## theta[706] 2.614978e-01 0.0003507930 0.01930304 2.246261e-01
## theta[707] 2.624154e-01 0.0008263936 0.04481128 1.788704e-01
## theta[708] 2.555080e-01 0.0003566035 0.01858390 2.200569e-01
## theta[709] 2.479273e-01 0.0008950586 0.04558454 1.645223e-01
## theta[710] 2.199465e-01 0.0004481408 0.02371732 1.760512e-01
## theta[711] 1.672818e-01 0.0009183123 0.04643687 8.551173e-02
## theta[712] 1.370920e-01 0.0008051386 0.04563661 6.023266e-02
## theta[713] 1.373540e-01 0.0006891961 0.03656285 7.805274e-02
## theta[714] 2.422597e-01 0.0003585643 0.01961849 2.059524e-01
## theta[715] 2.471365e-01 0.0006915163 0.03565143 1.829998e-01
## theta[716] 2.779953e-01 0.0003787180 0.02280425 2.353134e-01
## theta[717] 2.180008e-01 0.0009042691 0.04166039 1.425850e-01
## theta[718] 1.259019e-01 0.0008427006 0.04603664 5.187025e-02
## theta[719] 1.256415e-01 0.0009915797 0.04514583 4.665105e-02
## theta[720] 2.391793e-01 0.0004653833 0.02590816 1.919593e-01
## theta[721] 1.225252e-01 0.0009443905 0.04856310 4.233399e-02
## theta[722] 2.342012e-01 0.0003080509 0.01653336 2.018468e-01
## theta[723] 1.460563e-01 0.0006088773 0.03316050 8.634122e-02
## theta[724] 1.365314e-01 0.0010485667 0.04716866 5.805339e-02
## theta[725] 2.634099e-01 0.0005868363 0.03221934 2.051463e-01
## theta[726] 2.433111e-01 0.0007039396 0.03662257 1.757074e-01
## theta[727] 1.280239e-01 0.0010496158 0.04804075 5.021323e-02
## theta[728] 1.253140e-01 0.0009279364 0.04648524 4.968398e-02
## theta[729] 2.190487e-01 0.0003703415 0.01889066 1.828140e-01
## theta[730] 3.040350e-01 0.0003255504 0.01714455 2.704354e-01
## theta[731] 2.135890e-01 0.0003483404 0.02049843 1.743182e-01
## theta[732] 2.765276e-01 0.0004688240 0.02469345 2.290714e-01
## theta[733] 1.269047e-01 0.0009692117 0.04840058 4.604655e-02
## theta[734] 2.556669e-01 0.0004829047 0.02696651 2.056634e-01
## theta[735] 2.467079e-01 0.0006629946 0.03460712 1.843659e-01
## theta[736] 8.909902e-02 0.0006324295 0.02909552 3.787508e-02
## theta[737] 1.154457e-01 0.0008324008 0.04182113 4.918010e-02
## theta[738] 1.491553e-01 0.0006492053 0.03407364 9.063206e-02
## theta[739] 2.827288e-01 0.0003610727 0.01773879 2.480188e-01
## theta[740] 2.589127e-01 0.0005382547 0.03185116 1.988443e-01
## theta[741] 2.156324e-01 0.0002528291 0.01789409 1.811751e-01
## theta[742] 1.148429e-01 0.0006971264 0.03418818 5.779089e-02
## theta[743] 2.689685e-01 0.0003161614 0.01863493 2.337548e-01
## theta[744] 2.296995e-01 0.0004548355 0.02611594 1.806900e-01
## theta[745] 2.663280e-01 0.0003878107 0.02288403 2.204526e-01
## theta[746] 2.304806e-01 0.0003701758 0.02205188 1.882455e-01
## theta[747] 2.453298e-01 0.0006421145 0.03238358 1.841291e-01
## theta[748] 2.467521e-01 0.0008990333 0.04313718 1.670097e-01
## theta[749] 1.869420e-01 0.0005641181 0.02606897 1.385644e-01
## theta[750] 2.699458e-01 0.0003144590 0.01896718 2.322339e-01
## theta[751] 1.272544e-01 0.0009496062 0.04626745 4.700608e-02
## theta[752] 1.964490e-01 0.0004659461 0.02395685 1.504087e-01
## theta[753] 2.381937e-01 0.0003271681 0.01891985 2.022594e-01
## theta[754] 2.594854e-01 0.0004205749 0.02342571 2.142152e-01
## theta[755] 2.646065e-01 0.0003350082 0.02001560 2.282323e-01
## theta[756] 1.105745e-01 0.0005518159 0.03168029 5.449320e-02
## theta[757] 1.282222e-01 0.0008827091 0.04706888 5.027225e-02
## theta[758] 2.464986e-01 0.0006032364 0.03083728 1.870448e-01
## theta[759] 2.788558e-01 0.0003351465 0.01725424 2.450390e-01
## theta[760] 2.863576e-01 0.0004130034 0.02351101 2.423024e-01
## theta[761] 2.561182e-01 0.0005744350 0.03128886 1.969519e-01
## theta[762] 1.264862e-01 0.0008927918 0.04817649 4.896087e-02
## theta[763] 2.492215e-01 0.0008249555 0.04054215 1.752535e-01
## theta[764] 1.427389e-01 0.0009095799 0.04763551 6.233020e-02
## theta[765] 2.114300e-01 0.0005180122 0.03000038 1.536434e-01
## theta[766] 1.288890e-01 0.0008980156 0.04365239 5.343560e-02
## theta[767] 1.360472e-01 0.0008275429 0.04664409 5.674070e-02
## theta[768] 2.596579e-01 0.0003692891 0.02049912 2.208361e-01
## theta[769] 1.240056e-01 0.0009614398 0.04492645 4.861155e-02
## theta[770] 2.568466e-01 0.0004010517 0.02496596 2.099675e-01
## theta[771] 2.814252e-01 0.0004974054 0.02667100 2.320447e-01
## theta[772] 2.489734e-01 0.0003912322 0.02167601 2.089342e-01
## theta[773] 2.270919e-01 0.0005558132 0.03280024 1.663400e-01
## theta[774] 2.315235e-01 0.0008683213 0.04103469 1.573865e-01
## theta[775] 2.226471e-01 0.0003322684 0.02136814 1.818171e-01
## theta[776] 2.416081e-01 0.0003316416 0.01918669 2.067814e-01
## theta[777] 1.272246e-01 0.0009165267 0.04808973 4.734829e-02
## theta[778] 2.530190e-01 0.0004612439 0.02702589 2.018809e-01
## theta[779] 2.556872e-01 0.0003193184 0.01683285 2.243822e-01
## theta[780] 1.183282e-01 0.0006310897 0.03395174 6.044176e-02
## theta[781] 1.609014e-01 0.0011748387 0.05339837 7.456147e-02
## theta[782] 2.168053e-01 0.0003608298 0.02190477 1.737750e-01
## theta[783] 2.680823e-01 0.0004063091 0.02283419 2.235299e-01
## theta[784] 2.792294e-01 0.0003083206 0.01622300 2.493669e-01
## theta[785] 2.522614e-01 0.0004365816 0.02601028 2.038799e-01
## theta[786] 2.298611e-01 0.0004169294 0.02112064 1.902148e-01
## theta[787] 1.486023e-01 0.0008628448 0.04632319 7.550271e-02
## theta[788] 1.997066e-01 0.0009448624 0.04176203 1.264566e-01
## theta[789] 1.869751e-01 0.0008071006 0.03950994 1.149162e-01
## theta[790] 1.250006e-01 0.0008819242 0.04456079 4.868068e-02
## theta[791] 2.124539e-01 0.0004368824 0.02588789 1.627583e-01
## theta[792] 1.253788e-01 0.0009754621 0.04517089 5.413306e-02
## theta[793] 2.434094e-01 0.0004843844 0.02519638 1.971656e-01
## theta[794] 2.467365e-01 0.0006904106 0.04077828 1.761007e-01
## theta[795] 2.353158e-01 0.0007503836 0.04369387 1.535682e-01
## theta[796] 2.209060e-01 0.0006308699 0.03300472 1.612319e-01
## theta[797] 2.174982e-01 0.0004932985 0.02764404 1.659278e-01
## theta[798] 2.067410e-01 0.0006207536 0.03258258 1.432097e-01
## theta[799] 1.131855e-01 0.0008193809 0.03895700 4.747628e-02
## theta[800] 2.436909e-01 0.0006490208 0.04196181 1.659364e-01
## theta[801] 2.442828e-01 0.0007864573 0.04171036 1.681695e-01
## theta[802] 1.246147e-01 0.0010011651 0.04652130 4.934895e-02
## theta[803] 2.434681e-01 0.0007334571 0.04274378 1.646874e-01
## theta[804] 1.740940e-01 0.0007171535 0.03764031 1.085905e-01
## theta[805] 1.084718e-01 0.0005611720 0.03145795 5.467991e-02
## theta[806] 1.267211e-01 0.0009038071 0.04540423 5.337047e-02
## theta[807] 2.694864e-01 0.0003414709 0.01922229 2.325236e-01
## theta[808] 2.416770e-01 0.0004456806 0.02392550 1.972683e-01
## theta[809] 1.294453e-01 0.0009698650 0.04864686 5.025404e-02
## theta[810] 1.508568e-01 0.0008477374 0.04705875 7.357892e-02
## theta[811] 1.284490e-01 0.0008598775 0.04635770 4.994089e-02
## theta[812] 7.536781e-02 0.0007642380 0.02710271 2.937306e-02
## theta[813] 1.274694e-01 0.0008620369 0.04550894 5.412926e-02
## theta[814] 1.366892e-01 0.0009153373 0.04636638 6.071802e-02
## theta[815] 2.535985e-01 0.0006640481 0.03296822 1.905787e-01
## theta[816] 2.613606e-01 0.0005337578 0.02783167 2.112016e-01
## theta[817] 1.257772e-01 0.0009834808 0.04678865 5.036323e-02
## theta[818] 2.478659e-01 0.0005372835 0.02946472 1.939060e-01
## theta[819] 1.280973e-01 0.0010277983 0.04843979 4.914047e-02
## theta[820] 3.043066e-01 0.0002963850 0.01766352 2.710905e-01
## theta[821] 1.224637e-01 0.0008773428 0.04458461 4.994620e-02
## theta[822] 2.375602e-01 0.0005759843 0.03217943 1.758517e-01
## theta[823] 2.547358e-01 0.0004505404 0.02641210 2.051440e-01
## theta[824] 2.559435e-01 0.0004900713 0.02615424 2.080309e-01
## theta[825] 1.567556e-01 0.0006086708 0.03466146 9.790966e-02
## theta[826] 2.334822e-01 0.0005586893 0.02607620 1.825632e-01
## theta[827] 1.193365e-01 0.0008371472 0.04149092 4.994777e-02
## theta[828] 1.234866e-01 0.0005642112 0.03027282 7.031398e-02
## theta[829] 2.380241e-01 0.0002970410 0.01806281 2.044499e-01
## theta[830] 2.734728e-01 0.0005115958 0.02855085 2.207066e-01
## theta[831] 2.182416e-01 0.0007194010 0.03863547 1.474769e-01
## theta[832] 2.393999e-01 0.0003731189 0.01984841 2.009212e-01
## theta[833] 1.563321e-01 0.0007644500 0.04342454 8.146062e-02
## theta[834] 1.216808e-01 0.0007850766 0.04263240 5.157349e-02
## theta[835] 2.836133e-01 0.0003212438 0.01878314 2.471313e-01
## theta[836] 2.248789e-01 0.0004394936 0.02775763 1.729692e-01
## theta[837] 2.509005e-01 0.0009995071 0.04786701 1.605246e-01
## theta[838] 2.343058e-01 0.0003901409 0.02116755 1.946442e-01
## theta[839] 2.571519e-01 0.0005075087 0.02692462 2.085090e-01
## theta[840] 2.429746e-01 0.0005311472 0.02675456 1.945786e-01
## theta[841] 1.127272e-01 0.0009361931 0.04007493 4.488380e-02
## theta[842] 1.352359e-01 0.0007517770 0.04861219 5.588291e-02
## theta[843] 1.260307e-01 0.0009468898 0.04537563 4.790509e-02
## theta[844] 1.145467e-01 0.0007926049 0.04228680 4.443373e-02
## theta[845] 1.104805e-01 0.0006117504 0.03279514 5.409745e-02
## theta[846] 2.378622e-01 0.0004149194 0.02374566 1.927349e-01
## theta[847] 1.302560e-01 0.0009074105 0.04707917 5.408626e-02
## theta[848] 1.582674e-01 0.0009737534 0.04935148 7.815188e-02
## theta[849] 2.137610e-01 0.0005477602 0.03048665 1.575999e-01
## theta[850] 1.442651e-01 0.0010772744 0.05146805 6.186417e-02
## theta[851] 1.201088e-01 0.0008520148 0.04350780 4.958429e-02
## theta[852] 1.530075e-01 0.0006075013 0.03297798 9.414323e-02
## theta[853] 9.441733e-02 0.0005774151 0.02794365 4.653451e-02
## theta[854] 2.562841e-01 0.0005154236 0.02795293 2.031075e-01
## theta[855] 1.235811e-01 0.0009227910 0.04522552 4.652137e-02
## theta[856] 2.667487e-01 0.0006019632 0.03451640 2.016113e-01
## theta[857] 2.646457e-01 0.0003772649 0.01997785 2.272571e-01
## theta[858] 2.796220e-01 0.0004436193 0.02325404 2.348966e-01
## theta[859] 2.350161e-01 0.0004993106 0.02865479 1.813441e-01
## theta[860] 2.622716e-01 0.0003789558 0.01910677 2.258861e-01
## theta[861] 2.908960e-01 0.0003745948 0.02080478 2.487655e-01
## theta[862] 2.091592e-01 0.0007193775 0.03363994 1.450630e-01
## theta[863] 2.596169e-01 0.0003695936 0.02086975 2.197322e-01
## theta[864] 2.550649e-01 0.0006173725 0.03332419 1.920841e-01
## theta[865] 2.172892e-01 0.0004119818 0.02485504 1.706483e-01
## theta[866] 1.270785e-01 0.0008292092 0.04331693 5.219872e-02
## theta[867] 2.360292e-01 0.0005408895 0.03000850 1.818594e-01
## theta[868] 1.276137e-01 0.0011470232 0.05219572 4.160230e-02
## theta[869] 2.100058e-01 0.0004471806 0.02539846 1.623225e-01
## theta[870] 2.773296e-01 0.0006264639 0.03154429 2.177467e-01
## theta[871] 3.032966e-01 0.0003542757 0.01865153 2.658371e-01
## theta[872] 2.220784e-01 0.0006334601 0.03349817 1.606393e-01
## theta[873] 2.294973e-01 0.0005909587 0.02970325 1.729258e-01
## theta[874] 2.804589e-01 0.0005139648 0.02743873 2.291607e-01
## theta[875] 2.844677e-01 0.0003231073 0.01800145 2.499998e-01
## theta[876] 2.700681e-01 0.0003091229 0.01686924 2.369549e-01
## theta[877] 2.303761e-01 0.0004916687 0.02394479 1.849805e-01
## theta[878] 1.484332e-01 0.0005854842 0.03331812 8.981562e-02
## theta[879] 1.408331e-01 0.0009000346 0.04910957 5.874856e-02
## theta[880] 2.521361e-01 0.0003190382 0.01698711 2.196719e-01
## theta[881] 2.351660e-01 0.0004261652 0.02333405 1.900017e-01
## theta[882] 1.330466e-01 0.0009273911 0.04957244 5.226502e-02
## theta[883] 1.230513e-01 0.0010782103 0.04582143 4.916931e-02
## theta[884] 2.372052e-01 0.0006552010 0.04085176 1.605703e-01
## theta[885] 2.461293e-01 0.0002774149 0.01786037 2.115562e-01
## theta[886] 1.610969e-01 0.0008348030 0.04246214 9.102617e-02
## gamma[1] -1.091803e+00 0.0008130908 0.03512377 -1.161344e+00
## gamma[2] -1.095686e+00 0.0009696266 0.03916836 -1.172931e+00
## gamma[3] -1.069558e+00 0.0008062065 0.03511457 -1.135156e+00
## gamma[4] -1.169706e+00 0.0010500937 0.03689284 -1.241353e+00
## gamma[5] -1.049842e+00 0.0009340004 0.03996374 -1.130452e+00
## gamma[6] -1.104521e+00 0.0007643501 0.03177125 -1.166634e+00
## gamma[7] -1.908595e+00 0.0022988606 0.05639586 -2.016521e+00
## gamma[8] -1.052993e+00 0.0008091766 0.03685903 -1.121765e+00
## gamma[9] -1.102646e+00 0.0009402459 0.04179499 -1.186589e+00
## alpha_w 2.019431e-02 0.0004873033 0.01687514 -1.369954e-02
## alpha_h -9.317251e-03 0.0004958841 0.01641467 -4.258818e-02
## kappa[1] 1.082383e+02 0.6568029556 22.99712537 6.825202e+01
## kappa[2] 1.059715e+02 0.7499067462 23.74032693 6.533416e+01
## kappa[3] 9.657038e+01 0.6064782726 21.64684708 6.053540e+01
## kappa[4] 9.037510e+01 0.6501500333 18.39875993 5.895969e+01
## kappa[5] 9.388812e+01 0.5980293906 22.38752127 5.715590e+01
## kappa[6] 1.071271e+02 0.6875572139 22.14657895 7.086343e+01
## kappa[7] 5.235693e+01 1.0929918723 13.02868319 3.226598e+01
## kappa[8] 1.108521e+02 0.6309707285 26.35051072 6.615243e+01
## kappa[9] 8.922821e+01 0.6395735551 21.09917420 5.376222e+01
## lp__ -8.740610e+04 2.9763396362 37.65788030 -8.747895e+04
## 25% 50% 75% 97.5%
## theta[1] 1.931620e-01 2.137932e-01 2.350119e-01 2.718659e-01
## theta[2] 7.817197e-02 9.842435e-02 1.194538e-01 1.639960e-01
## theta[3] 2.832755e-01 2.954227e-01 3.068739e-01 3.292609e-01
## theta[4] 9.220098e-02 1.200623e-01 1.557910e-01 2.374361e-01
## theta[5] 9.325016e-02 1.226575e-01 1.551296e-01 2.313256e-01
## theta[6] 2.030085e-01 2.132549e-01 2.234132e-01 2.493983e-01
## theta[7] 2.722173e-01 2.831979e-01 2.937634e-01 3.164398e-01
## theta[8] 2.387245e-01 2.589630e-01 2.782913e-01 3.139844e-01
## theta[9] 2.563202e-01 2.679045e-01 2.793602e-01 3.022493e-01
## theta[10] 2.398651e-01 2.534864e-01 2.682564e-01 2.976219e-01
## theta[11] 2.010567e-01 2.281651e-01 2.568111e-01 3.200664e-01
## theta[12] 9.629665e-02 1.227075e-01 1.505539e-01 2.195724e-01
## theta[13] 2.142861e-01 2.350453e-01 2.550459e-01 2.995333e-01
## theta[14] 1.053596e-01 1.231979e-01 1.437762e-01 1.896626e-01
## theta[15] 2.338410e-01 2.529415e-01 2.728911e-01 3.136121e-01
## theta[16] 3.008441e-01 3.126815e-01 3.247502e-01 3.476761e-01
## theta[17] 2.025212e-01 2.244172e-01 2.465213e-01 2.971152e-01
## theta[18] 2.805585e-01 2.918132e-01 3.028257e-01 3.262762e-01
## theta[19] 2.210415e-01 2.443293e-01 2.698567e-01 3.174170e-01
## theta[20] 2.678857e-01 2.802330e-01 2.929986e-01 3.166571e-01
## theta[21] 2.403609e-01 2.537068e-01 2.655189e-01 2.919239e-01
## theta[22] 2.234945e-01 2.454773e-01 2.675102e-01 3.222171e-01
## theta[23] 2.759219e-01 2.863827e-01 2.985496e-01 3.189756e-01
## theta[24] 2.650237e-01 2.765831e-01 2.877249e-01 3.094551e-01
## theta[25] 2.280489e-01 2.411780e-01 2.544674e-01 2.794750e-01
## theta[26] 2.097388e-01 2.352532e-01 2.598398e-01 3.118281e-01
## theta[27] 2.764675e-01 2.882248e-01 2.997571e-01 3.215686e-01
## theta[28] 8.941342e-02 1.154049e-01 1.491642e-01 2.235345e-01
## theta[29] 2.185549e-01 2.376616e-01 2.596668e-01 2.986445e-01
## theta[30] 2.256459e-01 2.393836e-01 2.541587e-01 2.814038e-01
## theta[31] 2.840775e-01 2.963823e-01 3.083076e-01 3.296938e-01
## theta[32] 2.572604e-01 2.697017e-01 2.817256e-01 3.070063e-01
## theta[33] 1.186432e-01 1.438967e-01 1.705934e-01 2.393071e-01
## theta[34] 2.501978e-01 2.619907e-01 2.739568e-01 2.948368e-01
## theta[35] 2.269482e-01 2.417663e-01 2.570061e-01 2.888110e-01
## theta[36] 2.271176e-01 2.411894e-01 2.571294e-01 2.841153e-01
## theta[37] 1.054054e-01 1.350250e-01 1.714909e-01 2.490235e-01
## theta[38] 2.487448e-01 2.597048e-01 2.701221e-01 2.916389e-01
## theta[39] 9.061539e-02 1.164874e-01 1.473709e-01 2.197963e-01
## theta[40] 1.977406e-01 2.248005e-01 2.532908e-01 3.081435e-01
## theta[41] 2.837795e-01 2.971278e-01 3.095093e-01 3.331379e-01
## theta[42] 2.678370e-01 2.796602e-01 2.910023e-01 3.136418e-01
## theta[43] 2.545484e-01 2.744554e-01 2.933235e-01 3.345229e-01
## theta[44] 2.230393e-01 2.351993e-01 2.486187e-01 2.748060e-01
## theta[45] 1.002942e-01 1.270184e-01 1.584755e-01 2.294148e-01
## theta[46] 3.059067e-01 3.169068e-01 3.288684e-01 3.521255e-01
## theta[47] 2.460569e-01 2.727687e-01 3.020748e-01 3.678773e-01
## theta[48] 9.077880e-02 1.175308e-01 1.439376e-01 2.079555e-01
## theta[49] 1.989122e-01 2.142396e-01 2.289475e-01 2.588649e-01
## theta[50] 2.878626e-01 3.030385e-01 3.172850e-01 3.449744e-01
## theta[51] 2.043814e-01 2.222145e-01 2.429081e-01 2.824992e-01
## theta[52] 1.066853e-01 1.335789e-01 1.663649e-01 2.443265e-01
## theta[53] 1.146872e-01 1.443687e-01 1.807859e-01 2.724571e-01
## theta[54] 2.723308e-01 2.833057e-01 2.951048e-01 3.181619e-01
## theta[55] 9.238335e-02 1.158329e-01 1.416811e-01 1.969351e-01
## theta[56] 1.008877e-01 1.250582e-01 1.524845e-01 2.081772e-01
## theta[57] 2.186819e-01 2.352472e-01 2.520165e-01 2.866197e-01
## theta[58] 1.789724e-01 2.044248e-01 2.312721e-01 2.808126e-01
## theta[59] 2.622044e-01 2.767100e-01 2.910703e-01 3.198205e-01
## theta[60] 9.076224e-02 1.181263e-01 1.499494e-01 2.255721e-01
## theta[61] 2.815008e-01 2.929693e-01 3.051094e-01 3.306405e-01
## theta[62] 8.484694e-02 1.085270e-01 1.376206e-01 2.079572e-01
## theta[63] 2.563936e-01 2.670586e-01 2.789820e-01 3.032648e-01
## theta[64] 2.052619e-01 2.257383e-01 2.478751e-01 2.898258e-01
## theta[65] 2.820097e-01 2.947572e-01 3.059384e-01 3.299174e-01
## theta[66] 2.302401e-01 2.472881e-01 2.660907e-01 3.016514e-01
## theta[67] 2.549472e-01 2.786059e-01 3.034478e-01 3.582998e-01
## theta[68] 9.338922e-02 1.226384e-01 1.606463e-01 2.392454e-01
## theta[69] 8.065608e-02 9.960883e-02 1.196784e-01 1.663603e-01
## theta[70] 9.465711e-02 1.240228e-01 1.558233e-01 2.317455e-01
## theta[71] 2.285700e-01 2.450725e-01 2.622919e-01 2.969386e-01
## theta[72] 2.737877e-01 2.863413e-01 2.991453e-01 3.245164e-01
## theta[73] 7.258045e-02 9.653861e-02 1.248988e-01 1.861735e-01
## theta[74] 2.567654e-01 2.672035e-01 2.788401e-01 3.010692e-01
## theta[75] 2.931094e-01 3.048699e-01 3.165597e-01 3.387974e-01
## theta[76] 9.247746e-02 1.206003e-01 1.545620e-01 2.269121e-01
## theta[77] 1.979270e-01 2.230064e-01 2.503586e-01 2.987084e-01
## theta[78] 1.932938e-01 2.191923e-01 2.475722e-01 3.051584e-01
## theta[79] 2.022101e-01 2.301363e-01 2.564529e-01 3.146430e-01
## theta[80] 2.291547e-01 2.446005e-01 2.618328e-01 2.931884e-01
## theta[81] 2.019624e-01 2.194952e-01 2.388563e-01 2.774875e-01
## theta[82] 9.452490e-02 1.242835e-01 1.553857e-01 2.338515e-01
## theta[83] 9.512203e-02 1.239064e-01 1.548883e-01 2.370165e-01
## theta[84] 9.223980e-02 1.217962e-01 1.582867e-01 2.328647e-01
## theta[85] 1.091400e-01 1.374404e-01 1.724722e-01 2.409610e-01
## theta[86] 9.654165e-02 1.260997e-01 1.617969e-01 2.392959e-01
## theta[87] 1.035798e-01 1.315021e-01 1.660858e-01 2.314853e-01
## theta[88] 2.006845e-01 2.256433e-01 2.541824e-01 3.072550e-01
## theta[89] 2.081380e-01 2.278334e-01 2.494573e-01 2.911458e-01
## theta[90] 1.216233e-01 1.460470e-01 1.752303e-01 2.325701e-01
## theta[91] 2.566833e-01 2.702793e-01 2.844537e-01 3.129324e-01
## theta[92] 2.351708e-01 2.479615e-01 2.608269e-01 2.834338e-01
## theta[93] 1.056556e-01 1.357892e-01 1.677831e-01 2.457811e-01
## theta[94] 9.383875e-02 1.195460e-01 1.538098e-01 2.249497e-01
## theta[95] 2.110255e-01 2.374442e-01 2.657958e-01 3.237384e-01
## theta[96] 1.935221e-01 2.150581e-01 2.395369e-01 2.860564e-01
## theta[97] 2.118419e-01 2.255196e-01 2.396152e-01 2.678366e-01
## theta[98] 2.299376e-01 2.557937e-01 2.816501e-01 3.309850e-01
## theta[99] 2.637311e-01 2.793181e-01 2.948720e-01 3.249362e-01
## theta[100] 2.639831e-01 2.761665e-01 2.878866e-01 3.099600e-01
## theta[101] 2.360252e-01 2.598751e-01 2.849586e-01 3.346165e-01
## theta[102] 2.220044e-01 2.381031e-01 2.540796e-01 2.845898e-01
## theta[103] 1.899067e-01 2.025775e-01 2.150144e-01 2.395995e-01
## theta[104] 2.303354e-01 2.429882e-01 2.570154e-01 2.803648e-01
## theta[105] 1.985020e-01 2.142761e-01 2.293651e-01 2.624397e-01
## theta[106] 1.947028e-01 2.129198e-01 2.335417e-01 2.752294e-01
## theta[107] 2.220902e-01 2.530609e-01 2.861104e-01 3.460886e-01
## theta[108] 9.280975e-02 1.222379e-01 1.553701e-01 2.265908e-01
## theta[109] 2.390419e-01 2.620027e-01 2.881142e-01 3.469449e-01
## theta[110] 1.963583e-01 2.161945e-01 2.369303e-01 2.687117e-01
## theta[111] 1.928548e-01 2.109661e-01 2.310003e-01 2.686505e-01
## theta[112] 2.575611e-01 2.701668e-01 2.822050e-01 3.067622e-01
## theta[113] 2.089120e-01 2.284028e-01 2.491525e-01 2.906523e-01
## theta[114] 2.388132e-01 2.547252e-01 2.697849e-01 3.082574e-01
## theta[115] 2.074133e-01 2.279480e-01 2.487903e-01 2.893797e-01
## theta[116] 2.032722e-01 2.157410e-01 2.277762e-01 2.558261e-01
## theta[117] 2.221567e-01 2.375750e-01 2.525867e-01 2.823171e-01
## theta[118] 8.811842e-02 1.193505e-01 1.565174e-01 2.320899e-01
## theta[119] 1.021728e-01 1.287639e-01 1.646672e-01 2.356421e-01
## theta[120] 2.204894e-01 2.315684e-01 2.431309e-01 2.673093e-01
## theta[121] 1.984972e-01 2.205499e-01 2.437317e-01 2.850836e-01
## theta[122] 9.205427e-02 1.199641e-01 1.535292e-01 2.257732e-01
## theta[123] 2.104272e-01 2.304701e-01 2.519582e-01 2.929556e-01
## theta[124] 2.416145e-01 2.643051e-01 2.866872e-01 3.352419e-01
## theta[125] 1.121815e-01 1.332053e-01 1.593779e-01 2.082900e-01
## theta[126] 1.712053e-01 1.902765e-01 2.101657e-01 2.514787e-01
## theta[127] 1.050201e-01 1.345280e-01 1.680288e-01 2.422639e-01
## theta[128] 1.019934e-01 1.314061e-01 1.638807e-01 2.333611e-01
## theta[129] 2.089815e-01 2.343462e-01 2.641857e-01 3.213877e-01
## theta[130] 2.094039e-01 2.376834e-01 2.650632e-01 3.282399e-01
## theta[131] 2.442559e-01 2.576882e-01 2.696121e-01 2.965260e-01
## theta[132] 1.948199e-01 2.105306e-01 2.262854e-01 2.545198e-01
## theta[133] 2.570851e-01 2.681195e-01 2.799163e-01 3.034663e-01
## theta[134] 9.394054e-02 1.120396e-01 1.337414e-01 1.812928e-01
## theta[135] 8.778705e-02 1.202935e-01 1.528127e-01 2.257681e-01
## theta[136] 3.078079e-01 3.217087e-01 3.347292e-01 3.611167e-01
## theta[137] 2.323624e-01 2.441355e-01 2.562889e-01 2.796558e-01
## theta[138] 2.470655e-01 2.664649e-01 2.875896e-01 3.277344e-01
## theta[139] 9.451608e-02 1.197863e-01 1.567703e-01 2.264308e-01
## theta[140] 2.551009e-01 2.673725e-01 2.798184e-01 3.052721e-01
## theta[141] 2.278037e-01 2.522823e-01 2.793445e-01 3.276595e-01
## theta[142] 2.470147e-01 2.595508e-01 2.737879e-01 2.997317e-01
## theta[143] 2.820607e-01 2.938162e-01 3.051657e-01 3.303934e-01
## theta[144] 2.520008e-01 2.632273e-01 2.743248e-01 2.981396e-01
## theta[145] 2.130411e-01 2.340092e-01 2.567384e-01 3.028392e-01
## theta[146] 1.961717e-01 2.062143e-01 2.180890e-01 2.398154e-01
## theta[147] 2.419256e-01 2.571727e-01 2.739997e-01 3.043844e-01
## theta[148] 2.939053e-01 3.097095e-01 3.235447e-01 3.508870e-01
## theta[149] 2.388948e-01 2.504814e-01 2.621105e-01 2.861248e-01
## theta[150] 1.084355e-01 1.356347e-01 1.686454e-01 2.396818e-01
## theta[151] 2.177333e-01 2.440638e-01 2.726719e-01 3.288214e-01
## theta[152] 1.102258e-01 1.397255e-01 1.726368e-01 2.511241e-01
## theta[153] 1.235832e-01 1.485212e-01 1.771511e-01 2.409541e-01
## theta[154] 1.021309e-01 1.348638e-01 1.714672e-01 2.498663e-01
## theta[155] 1.097441e-01 1.371691e-01 1.681974e-01 2.452375e-01
## theta[156] 2.203472e-01 2.323896e-01 2.453859e-01 2.698567e-01
## theta[157] 2.124378e-01 2.255404e-01 2.392282e-01 2.698213e-01
## theta[158] 2.195225e-01 2.340986e-01 2.486027e-01 2.770733e-01
## theta[159] 9.251500e-02 1.195418e-01 1.526078e-01 2.192495e-01
## theta[160] 2.225272e-01 2.424388e-01 2.619663e-01 3.028723e-01
## theta[161] 9.188020e-02 1.207869e-01 1.512186e-01 2.298509e-01
## theta[162] 1.326360e-01 1.570536e-01 1.834925e-01 2.340821e-01
## theta[163] 7.836950e-02 1.046237e-01 1.362890e-01 2.053662e-01
## theta[164] 2.363819e-01 2.568044e-01 2.785277e-01 3.269941e-01
## theta[165] 2.505263e-01 2.700153e-01 2.890095e-01 3.293160e-01
## theta[166] 2.038547e-01 2.293839e-01 2.546122e-01 3.039252e-01
## theta[167] 2.691174e-01 2.812353e-01 2.930753e-01 3.153370e-01
## theta[168] 2.392080e-01 2.538909e-01 2.682822e-01 2.950298e-01
## theta[169] 2.672966e-01 2.799574e-01 2.946476e-01 3.194131e-01
## theta[170] 1.892821e-01 2.055738e-01 2.237315e-01 2.577985e-01
## theta[171] 8.905634e-02 1.145294e-01 1.439890e-01 2.213883e-01
## theta[172] 8.719113e-02 1.075724e-01 1.307619e-01 1.816366e-01
## theta[173] 1.750657e-01 1.956677e-01 2.156491e-01 2.611393e-01
## theta[174] 2.556766e-01 2.673272e-01 2.790208e-01 3.045218e-01
## theta[175] 2.697859e-01 2.833967e-01 2.977794e-01 3.241153e-01
## theta[176] 2.269733e-01 2.533091e-01 2.839358e-01 3.350666e-01
## theta[177] 2.479884e-01 2.645021e-01 2.809470e-01 3.171257e-01
## theta[178] 2.279093e-01 2.504041e-01 2.721905e-01 3.197071e-01
## theta[179] 2.491753e-01 2.619288e-01 2.746931e-01 3.019441e-01
## theta[180] 1.772053e-01 2.030911e-01 2.302765e-01 2.883427e-01
## theta[181] 2.235468e-01 2.396727e-01 2.563967e-01 2.866887e-01
## theta[182] 2.636623e-01 2.766809e-01 2.897066e-01 3.149922e-01
## theta[183] 9.447458e-02 1.216148e-01 1.541680e-01 2.309942e-01
## theta[184] 8.760657e-02 1.160025e-01 1.452589e-01 2.215651e-01
## theta[185] 2.764485e-01 2.915269e-01 3.070257e-01 3.350755e-01
## theta[186] 2.206721e-01 2.362555e-01 2.545491e-01 2.873063e-01
## theta[187] 9.017325e-02 1.182670e-01 1.472379e-01 2.256038e-01
## theta[188] 1.037911e-01 1.303712e-01 1.597272e-01 2.251875e-01
## theta[189] 8.698061e-02 1.146417e-01 1.473213e-01 2.245266e-01
## theta[190] 1.055403e-01 1.312943e-01 1.668263e-01 2.369382e-01
## theta[191] 1.790007e-01 1.937303e-01 2.090105e-01 2.393134e-01
## theta[192] 2.172739e-01 2.372264e-01 2.576123e-01 2.967583e-01
## theta[193] 1.160659e-01 1.424408e-01 1.717505e-01 2.284847e-01
## theta[194] 8.593959e-02 1.107394e-01 1.358024e-01 1.942811e-01
## theta[195] 8.813014e-02 1.163046e-01 1.511011e-01 2.290827e-01
## theta[196] 9.022586e-02 1.197483e-01 1.519354e-01 2.299545e-01
## theta[197] 1.456901e-01 1.692998e-01 1.950560e-01 2.465692e-01
## theta[198] 9.147535e-02 1.166447e-01 1.517671e-01 2.231862e-01
## theta[199] 8.670876e-02 1.044453e-01 1.220896e-01 1.641764e-01
## theta[200] 2.094098e-01 2.335310e-01 2.602233e-01 3.057252e-01
## theta[201] 2.087049e-01 2.207017e-01 2.325720e-01 2.584270e-01
## theta[202] 2.189903e-01 2.309485e-01 2.438309e-01 2.680876e-01
## theta[203] 2.466324e-01 2.584955e-01 2.717904e-01 2.973880e-01
## theta[204] 2.119654e-01 2.265037e-01 2.424394e-01 2.703813e-01
## theta[205] 2.538758e-01 2.646671e-01 2.756525e-01 2.990172e-01
## theta[206] 1.234122e-01 1.532044e-01 1.902107e-01 2.696919e-01
## theta[207] 2.160655e-01 2.269531e-01 2.385035e-01 2.612208e-01
## theta[208] 1.555738e-01 1.765361e-01 2.004337e-01 2.523875e-01
## theta[209] 2.351711e-01 2.577862e-01 2.809825e-01 3.256366e-01
## theta[210] 8.525786e-02 1.127564e-01 1.445721e-01 2.138891e-01
## theta[211] 2.066459e-01 2.317597e-01 2.606058e-01 3.145163e-01
## theta[212] 2.148053e-01 2.388314e-01 2.651314e-01 3.180618e-01
## theta[213] 2.127791e-01 2.365749e-01 2.612954e-01 3.063882e-01
## theta[214] 2.558617e-01 2.680226e-01 2.804683e-01 3.030977e-01
## theta[215] 9.397301e-02 1.215993e-01 1.511821e-01 2.267795e-01
## theta[216] 7.519191e-02 1.004504e-01 1.303961e-01 1.942912e-01
## theta[217] 8.784474e-02 1.146226e-01 1.458440e-01 2.209052e-01
## theta[218] 2.730471e-01 2.901168e-01 3.075151e-01 3.398273e-01
## theta[219] 8.532877e-02 1.144486e-01 1.442777e-01 2.138737e-01
## theta[220] 8.523712e-02 1.135281e-01 1.468973e-01 2.246562e-01
## theta[221] 2.243728e-01 2.353575e-01 2.453550e-01 2.646060e-01
## theta[222] 9.302775e-02 1.195738e-01 1.527277e-01 2.344389e-01
## theta[223] 9.217785e-02 1.157553e-01 1.434322e-01 1.989389e-01
## theta[224] 8.406641e-02 1.147968e-01 1.455355e-01 2.159138e-01
## theta[225] 9.407921e-02 1.204123e-01 1.514508e-01 2.188643e-01
## theta[226] 2.418275e-01 2.687867e-01 2.961535e-01 3.548537e-01
## theta[227] 9.199937e-02 1.203829e-01 1.569405e-01 2.265132e-01
## theta[228] 2.137129e-01 2.248597e-01 2.374821e-01 2.583087e-01
## theta[229] 9.263279e-02 1.210479e-01 1.513833e-01 2.310714e-01
## theta[230] 2.187170e-01 2.317054e-01 2.444574e-01 2.705438e-01
## theta[231] 1.033101e-01 1.318779e-01 1.623704e-01 2.295163e-01
## theta[232] 2.721001e-01 2.841355e-01 2.961487e-01 3.183674e-01
## theta[233] 2.302516e-01 2.445609e-01 2.591549e-01 2.861779e-01
## theta[234] 2.370293e-01 2.475210e-01 2.583283e-01 2.804861e-01
## theta[235] 1.939874e-01 2.127562e-01 2.297910e-01 2.674754e-01
## theta[236] 2.127097e-01 2.336173e-01 2.552934e-01 2.964492e-01
## theta[237] 2.080635e-01 2.257600e-01 2.457447e-01 2.828815e-01
## theta[238] 2.361613e-01 2.531455e-01 2.713168e-01 3.064025e-01
## theta[239] 2.441945e-01 2.718430e-01 2.979040e-01 3.541577e-01
## theta[240] 2.087272e-01 2.290721e-01 2.512946e-01 2.893646e-01
## theta[241] 2.416470e-01 2.528197e-01 2.634193e-01 2.855524e-01
## theta[242] 2.415666e-01 2.639142e-01 2.864091e-01 3.323276e-01
## theta[243] 2.597561e-01 2.790877e-01 2.990046e-01 3.370119e-01
## theta[244] 2.499306e-01 2.621758e-01 2.739744e-01 2.973596e-01
## theta[245] 2.750121e-01 2.863548e-01 2.981708e-01 3.232721e-01
## theta[246] 1.110719e-01 1.442483e-01 1.824447e-01 2.675328e-01
## theta[247] 9.142380e-02 1.209903e-01 1.532144e-01 2.253846e-01
## theta[248] 2.233643e-01 2.471553e-01 2.738102e-01 3.237158e-01
## theta[249] 2.807619e-01 2.940195e-01 3.071649e-01 3.333865e-01
## theta[250] 2.876174e-01 3.005716e-01 3.162840e-01 3.418894e-01
## theta[251] 9.012074e-02 1.151208e-01 1.447082e-01 2.182511e-01
## theta[252] 2.317861e-01 2.499527e-01 2.671340e-01 3.039499e-01
## theta[253] 2.868714e-01 2.988535e-01 3.114274e-01 3.359557e-01
## theta[254] 2.434682e-01 2.549998e-01 2.670150e-01 2.891488e-01
## theta[255] 2.170584e-01 2.307035e-01 2.437469e-01 2.719367e-01
## theta[256] 2.532831e-01 2.638470e-01 2.750212e-01 2.998949e-01
## theta[257] 2.669730e-01 2.791205e-01 2.915662e-01 3.145852e-01
## theta[258] 2.277996e-01 2.605212e-01 2.925358e-01 3.533525e-01
## theta[259] 8.977560e-02 1.173614e-01 1.513571e-01 2.307541e-01
## theta[260] 2.960922e-01 3.074185e-01 3.192165e-01 3.425683e-01
## theta[261] 1.229789e-01 1.526959e-01 1.860445e-01 2.654467e-01
## theta[262] 1.968941e-01 2.132028e-01 2.301687e-01 2.639403e-01
## theta[263] 2.342330e-01 2.446946e-01 2.575532e-01 2.828072e-01
## theta[264] 2.036053e-01 2.210783e-01 2.392616e-01 2.748725e-01
## theta[265] 2.388725e-01 2.540840e-01 2.712218e-01 3.041979e-01
## theta[266] 2.787889e-01 2.910708e-01 3.041513e-01 3.306859e-01
## theta[267] 2.256747e-01 2.544052e-01 2.847356e-01 3.513526e-01
## theta[268] 1.092963e-01 1.330410e-01 1.585870e-01 2.155868e-01
## theta[269] 2.397540e-01 2.624343e-01 2.841972e-01 3.314734e-01
## theta[270] 2.651899e-01 2.813420e-01 2.983753e-01 3.331077e-01
## theta[271] 2.269287e-01 2.428940e-01 2.603896e-01 2.965634e-01
## theta[272] 2.018374e-01 2.192248e-01 2.383799e-01 2.803508e-01
## theta[273] 2.127247e-01 2.328991e-01 2.522989e-01 2.928832e-01
## theta[274] 2.673577e-01 2.826564e-01 2.992651e-01 3.271340e-01
## theta[275] 8.865752e-02 1.163272e-01 1.492204e-01 2.297625e-01
## theta[276] 1.934174e-01 2.136906e-01 2.330108e-01 2.737050e-01
## theta[277] 9.204485e-02 1.191810e-01 1.515069e-01 2.249833e-01
## theta[278] 1.336051e-01 1.601180e-01 1.884754e-01 2.492364e-01
## theta[279] 9.070922e-02 1.194107e-01 1.535382e-01 2.341095e-01
## theta[280] 2.085358e-01 2.191987e-01 2.306999e-01 2.536158e-01
## theta[281] 2.297319e-01 2.485925e-01 2.683325e-01 3.150434e-01
## theta[282] 9.023998e-02 1.175421e-01 1.540213e-01 2.314327e-01
## theta[283] 2.183052e-01 2.293097e-01 2.408147e-01 2.615093e-01
## theta[284] 9.288607e-02 1.213987e-01 1.541459e-01 2.261104e-01
## theta[285] 2.702207e-01 2.822700e-01 2.951614e-01 3.155739e-01
## theta[286] 7.898371e-02 9.692345e-02 1.172774e-01 1.597955e-01
## theta[287] 2.095053e-01 2.294318e-01 2.489789e-01 2.883053e-01
## theta[288] 2.456956e-01 2.683675e-01 2.931503e-01 3.423552e-01
## theta[289] 9.115686e-02 1.188249e-01 1.529277e-01 2.356938e-01
## theta[290] 2.635258e-01 2.755429e-01 2.874056e-01 3.110139e-01
## theta[291] 1.567806e-01 1.794197e-01 2.059405e-01 2.556036e-01
## theta[292] 2.230914e-01 2.441283e-01 2.695317e-01 3.159672e-01
## theta[293] 1.927735e-01 2.204968e-01 2.487880e-01 3.135385e-01
## theta[294] 2.331706e-01 2.500398e-01 2.676948e-01 3.055845e-01
## theta[295] 2.280495e-01 2.430068e-01 2.582032e-01 2.887903e-01
## theta[296] 9.148387e-02 1.212161e-01 1.571810e-01 2.434169e-01
## theta[297] 2.695833e-01 2.811597e-01 2.919023e-01 3.158691e-01
## theta[298] 2.416977e-01 2.589600e-01 2.754995e-01 3.063569e-01
## theta[299] 2.090576e-01 2.279107e-01 2.464335e-01 2.788383e-01
## theta[300] 2.016327e-01 2.180887e-01 2.346305e-01 2.724899e-01
## theta[301] 2.579097e-01 2.741727e-01 2.914589e-01 3.211936e-01
## theta[302] 2.130026e-01 2.388715e-01 2.653695e-01 3.189386e-01
## theta[303] 2.193460e-01 2.454464e-01 2.753172e-01 3.301792e-01
## theta[304] 9.020935e-02 1.190629e-01 1.513904e-01 2.244943e-01
## theta[305] 2.032643e-01 2.228038e-01 2.414255e-01 2.742235e-01
## theta[306] 2.233529e-01 2.349362e-01 2.459902e-01 2.705939e-01
## theta[307] 1.843947e-01 2.035256e-01 2.239024e-01 2.640703e-01
## theta[308] 1.416852e-01 1.703982e-01 2.017885e-01 2.680476e-01
## theta[309] 2.234277e-01 2.382642e-01 2.519727e-01 2.826622e-01
## theta[310] 2.711380e-01 2.839178e-01 2.972167e-01 3.224687e-01
## theta[311] 9.475190e-02 1.165709e-01 1.416357e-01 1.947073e-01
## theta[312] 2.181495e-01 2.405807e-01 2.648289e-01 3.120000e-01
## theta[313] 2.236521e-01 2.442706e-01 2.647148e-01 3.036979e-01
## theta[314] 8.082750e-02 1.102204e-01 1.412385e-01 2.181586e-01
## theta[315] 1.049986e-01 1.330824e-01 1.655126e-01 2.500439e-01
## theta[316] 2.225815e-01 2.523660e-01 2.822806e-01 3.426115e-01
## theta[317] 2.640976e-01 2.782431e-01 2.948824e-01 3.266845e-01
## theta[318] 2.334223e-01 2.465674e-01 2.597839e-01 2.848785e-01
## theta[319] 2.402571e-01 2.596332e-01 2.783461e-01 3.154872e-01
## theta[320] 9.638159e-02 1.231275e-01 1.572109e-01 2.270290e-01
## theta[321] 1.061416e-01 1.358201e-01 1.683426e-01 2.447646e-01
## theta[322] 1.027266e-01 1.295322e-01 1.587586e-01 2.265369e-01
## theta[323] 2.241957e-01 2.418120e-01 2.600766e-01 3.008690e-01
## theta[324] 9.019873e-02 1.200245e-01 1.511863e-01 2.247144e-01
## theta[325] 9.061072e-02 1.200382e-01 1.568539e-01 2.340805e-01
## theta[326] 2.019936e-01 2.334426e-01 2.637225e-01 3.259394e-01
## theta[327] 9.127036e-02 1.186674e-01 1.505798e-01 2.199078e-01
## theta[328] 2.129742e-01 2.376824e-01 2.639452e-01 3.170783e-01
## theta[329] 2.386097e-01 2.572900e-01 2.762640e-01 3.123991e-01
## theta[330] 1.066727e-01 1.366881e-01 1.763135e-01 2.601472e-01
## theta[331] 2.455214e-01 2.569433e-01 2.696856e-01 2.926630e-01
## theta[332] 2.189978e-01 2.354691e-01 2.515725e-01 2.829530e-01
## theta[333] 2.153189e-01 2.278451e-01 2.400680e-01 2.663428e-01
## theta[334] 2.570055e-01 2.692608e-01 2.818043e-01 3.073061e-01
## theta[335] 9.032940e-02 1.195593e-01 1.527865e-01 2.302655e-01
## theta[336] 9.336448e-02 1.249229e-01 1.595623e-01 2.249443e-01
## theta[337] 1.999206e-01 2.260609e-01 2.557790e-01 3.100475e-01
## theta[338] 9.173537e-02 1.205943e-01 1.550013e-01 2.258322e-01
## theta[339] 2.021289e-01 2.186855e-01 2.377670e-01 2.715212e-01
## theta[340] 1.929453e-01 2.065994e-01 2.202629e-01 2.483110e-01
## theta[341] 2.485697e-01 2.668962e-01 2.862636e-01 3.256212e-01
## theta[342] 2.566665e-01 2.695376e-01 2.832921e-01 3.107284e-01
## theta[343] 2.714378e-01 2.841054e-01 2.961649e-01 3.188216e-01
## theta[344] 2.167268e-01 2.418079e-01 2.677352e-01 3.166465e-01
## theta[345] 8.763393e-02 1.063038e-01 1.268314e-01 1.736631e-01
## theta[346] 2.246005e-01 2.362033e-01 2.474771e-01 2.700233e-01
## theta[347] 1.971331e-01 2.148486e-01 2.333245e-01 2.676815e-01
## theta[348] 9.339976e-02 1.189760e-01 1.521562e-01 2.270574e-01
## theta[349] 2.633623e-01 2.775503e-01 2.926724e-01 3.215886e-01
## theta[350] 2.327227e-01 2.462069e-01 2.590861e-01 2.837513e-01
## theta[351] 1.072782e-01 1.346307e-01 1.628433e-01 2.332386e-01
## theta[352] 2.027521e-01 2.294212e-01 2.622915e-01 3.277006e-01
## theta[353] 1.977151e-01 2.244615e-01 2.511686e-01 3.108637e-01
## theta[354] 2.465432e-01 2.567099e-01 2.681059e-01 2.867401e-01
## theta[355] 2.337729e-01 2.665355e-01 2.977412e-01 3.649248e-01
## theta[356] 2.157820e-01 2.374930e-01 2.605874e-01 3.059024e-01
## theta[357] 1.217734e-01 1.512425e-01 1.849362e-01 2.646701e-01
## theta[358] 2.493567e-01 2.679835e-01 2.861361e-01 3.201742e-01
## theta[359] 9.374373e-02 1.199807e-01 1.547707e-01 2.233833e-01
## theta[360] 1.938562e-01 2.173133e-01 2.406087e-01 2.870794e-01
## theta[361] 2.263900e-01 2.544924e-01 2.864186e-01 3.439884e-01
## theta[362] 1.801393e-01 1.994761e-01 2.186840e-01 2.632774e-01
## theta[363] 9.214389e-02 1.187552e-01 1.502277e-01 2.235337e-01
## theta[364] 9.968886e-02 1.179181e-01 1.390370e-01 1.857382e-01
## theta[365] 2.704926e-01 2.820007e-01 2.935887e-01 3.147528e-01
## theta[366] 2.172126e-01 2.354713e-01 2.531109e-01 2.871477e-01
## theta[367] 2.412715e-01 2.535315e-01 2.647943e-01 2.866281e-01
## theta[368] 2.736401e-01 2.859793e-01 2.979333e-01 3.196486e-01
## theta[369] 8.177758e-02 1.011286e-01 1.217485e-01 1.673485e-01
## theta[370] 2.439908e-01 2.552672e-01 2.653667e-01 2.852439e-01
## theta[371] 2.047276e-01 2.214512e-01 2.384256e-01 2.711227e-01
## theta[372] 2.678858e-01 2.779030e-01 2.895982e-01 3.096526e-01
## theta[373] 2.686983e-01 2.907692e-01 3.126236e-01 3.593800e-01
## theta[374] 2.176675e-01 2.416637e-01 2.653544e-01 3.141696e-01
## theta[375] 1.010634e-01 1.292029e-01 1.623715e-01 2.391144e-01
## theta[376] 2.308491e-01 2.466349e-01 2.608048e-01 2.905814e-01
## theta[377] 7.370827e-02 9.517547e-02 1.219345e-01 1.804884e-01
## theta[378] 2.528287e-01 2.671366e-01 2.815440e-01 3.106351e-01
## theta[379] 1.525338e-01 1.803042e-01 2.089898e-01 2.717292e-01
## theta[380] 9.155509e-02 1.159483e-01 1.421271e-01 2.003224e-01
## theta[381] 8.978418e-02 1.154506e-01 1.486896e-01 2.302874e-01
## theta[382] 1.999942e-01 2.221061e-01 2.442927e-01 2.909708e-01
## theta[383] 8.708384e-02 1.132341e-01 1.432149e-01 2.071320e-01
## theta[384] 1.034662e-01 1.228557e-01 1.448853e-01 1.933718e-01
## theta[385] 2.181654e-01 2.438161e-01 2.725738e-01 3.320477e-01
## theta[386] 2.364615e-01 2.492286e-01 2.618985e-01 2.848256e-01
## theta[387] 1.123350e-01 1.349683e-01 1.560648e-01 2.090924e-01
## theta[388] 9.039149e-02 1.204204e-01 1.533645e-01 2.299685e-01
## theta[389] 1.015708e-01 1.299590e-01 1.640877e-01 2.353271e-01
## theta[390] 2.517242e-01 2.643962e-01 2.763048e-01 3.003380e-01
## theta[391] 9.362610e-02 1.219516e-01 1.535790e-01 2.309187e-01
## theta[392] 1.006900e-01 1.268730e-01 1.582915e-01 2.351888e-01
## theta[393] 9.213102e-02 1.236755e-01 1.527676e-01 2.367791e-01
## theta[394] 2.433204e-01 2.586013e-01 2.733498e-01 3.015559e-01
## theta[395] 8.254528e-02 1.088304e-01 1.410425e-01 2.132009e-01
## theta[396] 2.129120e-01 2.244851e-01 2.372991e-01 2.613618e-01
## theta[397] 1.694322e-01 1.887085e-01 2.118806e-01 2.558437e-01
## theta[398] 1.782088e-01 1.935953e-01 2.085730e-01 2.381659e-01
## theta[399] 8.694125e-02 1.131193e-01 1.448151e-01 2.173558e-01
## theta[400] 2.362026e-01 2.592153e-01 2.815466e-01 3.271579e-01
## theta[401] 2.351704e-01 2.516068e-01 2.675302e-01 2.990756e-01
## theta[402] 2.038587e-01 2.231113e-01 2.424080e-01 2.836197e-01
## theta[403] 2.186964e-01 2.406171e-01 2.633953e-01 3.023056e-01
## theta[404] 8.637141e-02 1.134330e-01 1.444817e-01 2.116074e-01
## theta[405] 2.569353e-01 2.678521e-01 2.790169e-01 3.036831e-01
## theta[406] 2.448847e-01 2.565140e-01 2.675516e-01 2.868642e-01
## theta[407] 2.414895e-01 2.530543e-01 2.636573e-01 2.887657e-01
## theta[408] 1.566850e-01 1.855707e-01 2.151382e-01 2.845766e-01
## theta[409] 2.307592e-01 2.608370e-01 2.924956e-01 3.513085e-01
## theta[410] 2.116699e-01 2.397640e-01 2.688198e-01 3.334266e-01
## theta[411] 1.213488e-01 1.515427e-01 1.835561e-01 2.657019e-01
## theta[412] 2.402380e-01 2.519230e-01 2.644281e-01 2.863062e-01
## theta[413] 2.300847e-01 2.469314e-01 2.636527e-01 2.987038e-01
## theta[414] 2.738615e-01 2.886517e-01 3.049138e-01 3.316572e-01
## theta[415] 2.358799e-01 2.553257e-01 2.736792e-01 3.109731e-01
## theta[416] 1.030226e-01 1.339593e-01 1.652024e-01 2.442192e-01
## theta[417] 2.289015e-01 2.430969e-01 2.573113e-01 2.836863e-01
## theta[418] 2.306700e-01 2.468839e-01 2.633544e-01 2.959143e-01
## theta[419] 1.963747e-01 2.190155e-01 2.429479e-01 2.944795e-01
## theta[420] 2.091596e-01 2.329125e-01 2.565114e-01 3.088710e-01
## theta[421] 8.109290e-02 1.006804e-01 1.247248e-01 1.722447e-01
## theta[422] 2.271573e-01 2.378597e-01 2.493813e-01 2.715803e-01
## theta[423] 9.366175e-02 1.174699e-01 1.405926e-01 1.972770e-01
## theta[424] 2.954949e-01 3.094559e-01 3.250929e-01 3.529773e-01
## theta[425] 9.029701e-02 1.175703e-01 1.486920e-01 2.146079e-01
## theta[426] 2.077242e-01 2.228929e-01 2.392788e-01 2.725945e-01
## theta[427] 9.076574e-02 1.118831e-01 1.355463e-01 1.858160e-01
## theta[428] 9.426829e-02 1.215887e-01 1.527209e-01 2.242355e-01
## theta[429] 2.130645e-01 2.255018e-01 2.372573e-01 2.603832e-01
## theta[430] 9.031099e-02 1.173355e-01 1.478204e-01 2.209316e-01
## theta[431] 1.077962e-01 1.377462e-01 1.696071e-01 2.480873e-01
## theta[432] 9.500250e-02 1.241502e-01 1.595957e-01 2.317185e-01
## theta[433] 8.783120e-02 1.097411e-01 1.334863e-01 1.936793e-01
## theta[434] 9.472421e-02 1.156226e-01 1.417758e-01 2.030489e-01
## theta[435] 8.750227e-02 1.170161e-01 1.526161e-01 2.343904e-01
## theta[436] 2.295427e-01 2.561636e-01 2.816514e-01 3.388250e-01
## theta[437] 1.070936e-01 1.364148e-01 1.721152e-01 2.479926e-01
## theta[438] 2.098140e-01 2.348809e-01 2.614391e-01 3.074332e-01
## theta[439] 8.684327e-02 1.150459e-01 1.488667e-01 2.180401e-01
## theta[440] 2.042164e-01 2.181343e-01 2.332613e-01 2.628156e-01
## theta[441] 2.319308e-01 2.468360e-01 2.621443e-01 2.895934e-01
## theta[442] 2.447711e-01 2.568640e-01 2.698840e-01 2.914945e-01
## theta[443] 9.193899e-02 1.225350e-01 1.544812e-01 2.381592e-01
## theta[444] 2.317675e-01 2.581295e-01 2.863880e-01 3.445627e-01
## theta[445] 2.297573e-01 2.405881e-01 2.518205e-01 2.760047e-01
## theta[446] 1.511011e-01 1.798868e-01 2.104812e-01 2.823604e-01
## theta[447] 2.350758e-01 2.522305e-01 2.702777e-01 3.035450e-01
## theta[448] 2.084401e-01 2.288460e-01 2.482161e-01 2.882070e-01
## theta[449] 2.392461e-01 2.494412e-01 2.603840e-01 2.828923e-01
## theta[450] 2.504250e-01 2.639686e-01 2.786180e-01 3.097307e-01
## theta[451] 7.974440e-02 9.813933e-02 1.204810e-01 1.616449e-01
## theta[452] 2.163100e-01 2.445429e-01 2.743583e-01 3.295123e-01
## theta[453] 2.064447e-01 2.223606e-01 2.388541e-01 2.735733e-01
## theta[454] 2.947588e-01 3.070896e-01 3.204202e-01 3.432741e-01
## theta[455] 8.606495e-02 1.086510e-01 1.317829e-01 1.845360e-01
## theta[456] 9.140811e-02 1.218464e-01 1.537616e-01 2.419734e-01
## theta[457] 1.121390e-01 1.428576e-01 1.759800e-01 2.506430e-01
## theta[458] 9.573891e-02 1.251545e-01 1.618444e-01 2.396497e-01
## theta[459] 3.036282e-01 3.186812e-01 3.332757e-01 3.605532e-01
## theta[460] 8.668129e-02 1.088619e-01 1.347894e-01 1.827282e-01
## theta[461] 9.009540e-02 1.198193e-01 1.534217e-01 2.319121e-01
## theta[462] 2.217591e-01 2.375803e-01 2.541589e-01 2.893916e-01
## theta[463] 1.896076e-01 2.018666e-01 2.150880e-01 2.419482e-01
## theta[464] 9.122352e-02 1.166622e-01 1.496210e-01 2.183596e-01
## theta[465] 2.029619e-01 2.247090e-01 2.456982e-01 2.875869e-01
## theta[466] 2.501629e-01 2.663636e-01 2.816660e-01 3.139089e-01
## theta[467] 9.625349e-02 1.235277e-01 1.549404e-01 2.261846e-01
## theta[468] 8.721485e-02 1.056345e-01 1.250701e-01 1.716387e-01
## theta[469] 2.154547e-01 2.300466e-01 2.453095e-01 2.809764e-01
## theta[470] 2.234869e-01 2.413956e-01 2.584705e-01 2.922487e-01
## theta[471] 2.821753e-01 2.958331e-01 3.102146e-01 3.347728e-01
## theta[472] 8.588167e-02 1.148397e-01 1.471267e-01 2.174667e-01
## theta[473] 2.407005e-01 2.576940e-01 2.757295e-01 3.083925e-01
## theta[474] 2.853037e-01 3.005960e-01 3.153883e-01 3.437355e-01
## theta[475] 8.681051e-02 1.128268e-01 1.435424e-01 2.094075e-01
## theta[476] 1.506989e-01 1.748943e-01 2.014567e-01 2.548467e-01
## theta[477] 2.430530e-01 2.574072e-01 2.731012e-01 3.029472e-01
## theta[478] 2.148308e-01 2.368787e-01 2.608272e-01 3.026373e-01
## theta[479] 1.165693e-01 1.377715e-01 1.634700e-01 2.244314e-01
## theta[480] 2.868890e-01 3.002067e-01 3.137320e-01 3.365038e-01
## theta[481] 2.079910e-01 2.210219e-01 2.344773e-01 2.627614e-01
## theta[482] 1.366063e-01 1.611884e-01 1.873571e-01 2.424316e-01
## theta[483] 2.260603e-01 2.413733e-01 2.583424e-01 2.937856e-01
## theta[484] 2.078683e-01 2.305751e-01 2.553845e-01 3.074787e-01
## theta[485] 1.191970e-01 1.491530e-01 1.839393e-01 2.611392e-01
## theta[486] 2.725592e-01 2.828660e-01 2.944416e-01 3.195809e-01
## theta[487] 2.276954e-01 2.480591e-01 2.672583e-01 3.120672e-01
## theta[488] 1.662766e-01 1.875306e-01 2.106057e-01 2.535090e-01
## theta[489] 2.083741e-01 2.258306e-01 2.429341e-01 2.808022e-01
## theta[490] 2.317157e-01 2.470676e-01 2.634744e-01 2.957781e-01
## theta[491] 9.859709e-02 1.284682e-01 1.663156e-01 2.409204e-01
## theta[492] 2.142772e-01 2.286545e-01 2.432385e-01 2.741758e-01
## theta[493] 8.872395e-02 1.165466e-01 1.528065e-01 2.291930e-01
## theta[494] 2.335540e-01 2.469673e-01 2.612183e-01 2.851611e-01
## theta[495] 7.986862e-02 1.018164e-01 1.269322e-01 1.845916e-01
## theta[496] 8.654420e-02 1.112575e-01 1.400556e-01 2.015083e-01
## theta[497] 2.300410e-01 2.444903e-01 2.581035e-01 2.863388e-01
## theta[498] 2.691776e-01 2.813737e-01 2.940704e-01 3.183023e-01
## theta[499] 2.222446e-01 2.381995e-01 2.532097e-01 2.805209e-01
## theta[500] 9.131588e-02 1.095578e-01 1.299975e-01 1.772421e-01
## theta[501] 9.033408e-02 1.156688e-01 1.427433e-01 2.070867e-01
## theta[502] 2.330650e-01 2.436338e-01 2.541505e-01 2.755432e-01
## theta[503] 9.192294e-02 1.175063e-01 1.453129e-01 2.156874e-01
## theta[504] 2.514356e-01 2.668581e-01 2.823530e-01 3.154061e-01
## theta[505] 9.254386e-02 1.229414e-01 1.539679e-01 2.380057e-01
## theta[506] 2.190777e-01 2.335121e-01 2.494236e-01 2.730042e-01
## theta[507] 1.174262e-01 1.460790e-01 1.809200e-01 2.579375e-01
## theta[508] 1.636090e-01 1.833944e-01 2.047804e-01 2.450325e-01
## theta[509] 2.461216e-01 2.582556e-01 2.710605e-01 2.962127e-01
## theta[510] 2.260273e-01 2.428751e-01 2.600132e-01 2.916608e-01
## theta[511] 1.000567e-01 1.248741e-01 1.535581e-01 2.069195e-01
## theta[512] 2.808053e-01 2.943150e-01 3.073042e-01 3.350106e-01
## theta[513] 2.306979e-01 2.455924e-01 2.611728e-01 2.877745e-01
## theta[514] 1.981765e-01 2.161721e-01 2.367852e-01 2.699950e-01
## theta[515] 9.270638e-02 1.211632e-01 1.543200e-01 2.303954e-01
## theta[516] 2.105582e-01 2.356817e-01 2.658558e-01 3.193165e-01
## theta[517] 1.808864e-01 2.028705e-01 2.254962e-01 2.699506e-01
## theta[518] 8.066959e-02 1.057085e-01 1.336821e-01 2.006198e-01
## theta[519] 9.112861e-02 1.197952e-01 1.554260e-01 2.330818e-01
## theta[520] 9.496230e-02 1.221802e-01 1.559327e-01 2.310966e-01
## theta[521] 2.216698e-01 2.347165e-01 2.481980e-01 2.733459e-01
## theta[522] 2.515746e-01 2.637312e-01 2.766259e-01 2.988865e-01
## theta[523] 2.835882e-01 2.992864e-01 3.152616e-01 3.451413e-01
## theta[524] 2.065940e-01 2.296964e-01 2.541579e-01 3.105450e-01
## theta[525] 2.099934e-01 2.223477e-01 2.358933e-01 2.604532e-01
## theta[526] 2.421540e-01 2.605942e-01 2.801999e-01 3.184350e-01
## theta[527] 2.644632e-01 2.755361e-01 2.876844e-01 3.095291e-01
## theta[528] 8.707310e-02 1.162293e-01 1.481088e-01 2.294830e-01
## theta[529] 2.175150e-01 2.286633e-01 2.390731e-01 2.608422e-01
## theta[530] 2.181384e-01 2.339461e-01 2.510087e-01 2.862625e-01
## theta[531] 2.566688e-01 2.682580e-01 2.804097e-01 3.056316e-01
## theta[532] 8.947044e-02 1.176904e-01 1.533001e-01 2.355887e-01
## theta[533] 9.538513e-02 1.141508e-01 1.368803e-01 1.840587e-01
## theta[534] 2.920697e-01 3.090898e-01 3.274231e-01 3.653453e-01
## theta[535] 2.061623e-01 2.333628e-01 2.626463e-01 3.227508e-01
## theta[536] 8.903130e-02 1.150851e-01 1.430621e-01 2.148043e-01
## theta[537] 2.266416e-01 2.387978e-01 2.515988e-01 2.781140e-01
## theta[538] 8.806301e-02 1.160589e-01 1.463163e-01 2.086580e-01
## theta[539] 2.366148e-01 2.523179e-01 2.699512e-01 2.980767e-01
## theta[540] 2.209863e-01 2.450411e-01 2.720436e-01 3.233668e-01
## theta[541] 1.949316e-01 2.235074e-01 2.523153e-01 3.146006e-01
## theta[542] 2.012088e-01 2.245781e-01 2.498706e-01 2.977080e-01
## theta[543] 1.928539e-01 2.180708e-01 2.426259e-01 3.022719e-01
## theta[544] 9.971132e-02 1.212707e-01 1.454464e-01 2.025706e-01
## theta[545] 2.243271e-01 2.530746e-01 2.839292e-01 3.427598e-01
## theta[546] 2.223481e-01 2.361464e-01 2.480849e-01 2.735036e-01
## theta[547] 2.262517e-01 2.533171e-01 2.831699e-01 3.412031e-01
## theta[548] 9.057417e-02 1.179138e-01 1.524570e-01 2.212566e-01
## theta[549] 1.077801e-01 1.308538e-01 1.565809e-01 2.067178e-01
## theta[550] 8.821694e-02 1.064174e-01 1.277268e-01 1.713870e-01
## theta[551] 9.241206e-02 1.216411e-01 1.562950e-01 2.369166e-01
## theta[552] 8.401117e-02 1.115909e-01 1.450699e-01 2.167306e-01
## theta[553] 2.478793e-01 2.589820e-01 2.706907e-01 2.926355e-01
## theta[554] 8.483794e-02 1.107276e-01 1.395341e-01 2.048325e-01
## theta[555] 2.341263e-01 2.539034e-01 2.752520e-01 3.179230e-01
## theta[556] 7.662129e-02 9.583036e-02 1.180830e-01 1.650026e-01
## theta[557] 2.133691e-01 2.391489e-01 2.662090e-01 3.238066e-01
## theta[558] 2.294589e-01 2.477525e-01 2.669260e-01 3.082023e-01
## theta[559] 2.039833e-01 2.259228e-01 2.496270e-01 3.009173e-01
## theta[560] 2.020447e-01 2.217520e-01 2.419064e-01 2.876870e-01
## theta[561] 9.156896e-02 1.202166e-01 1.507646e-01 2.273730e-01
## theta[562] 9.118151e-02 1.180512e-01 1.494071e-01 2.290859e-01
## theta[563] 2.520994e-01 2.665366e-01 2.807868e-01 3.079050e-01
## theta[564] 2.229560e-01 2.360873e-01 2.484790e-01 2.782574e-01
## theta[565] 2.381642e-01 2.662576e-01 2.939145e-01 3.471880e-01
## theta[566] 2.408801e-01 2.598122e-01 2.791104e-01 3.195404e-01
## theta[567] 2.458519e-01 2.627159e-01 2.811639e-01 3.126586e-01
## theta[568] 2.121756e-01 2.281101e-01 2.441753e-01 2.751409e-01
## theta[569] 2.316695e-01 2.445226e-01 2.578652e-01 2.846566e-01
## theta[570] 1.174607e-01 1.390085e-01 1.618994e-01 2.125540e-01
## theta[571] 1.070183e-01 1.356568e-01 1.695803e-01 2.436651e-01
## theta[572] 2.710336e-01 2.865229e-01 3.017080e-01 3.310702e-01
## theta[573] 1.768527e-01 2.036147e-01 2.309618e-01 2.886123e-01
## theta[574] 2.158272e-01 2.474199e-01 2.807263e-01 3.441937e-01
## theta[575] 9.180628e-02 1.213053e-01 1.549672e-01 2.276632e-01
## theta[576] 9.478414e-02 1.220571e-01 1.546881e-01 2.260981e-01
## theta[577] 2.143359e-01 2.288369e-01 2.433544e-01 2.751721e-01
## theta[578] 2.019560e-01 2.329579e-01 2.605213e-01 3.207723e-01
## theta[579] 9.276309e-02 1.213738e-01 1.560947e-01 2.245191e-01
## theta[580] 1.010094e-01 1.299854e-01 1.623149e-01 2.392733e-01
## theta[581] 2.038846e-01 2.302587e-01 2.579332e-01 3.116662e-01
## theta[582] 2.217042e-01 2.347781e-01 2.492991e-01 2.784350e-01
## theta[583] 2.353539e-01 2.556931e-01 2.764070e-01 3.177744e-01
## theta[584] 1.250082e-01 1.477609e-01 1.711456e-01 2.188971e-01
## theta[585] 2.402659e-01 2.558520e-01 2.711059e-01 2.989181e-01
## theta[586] 9.864458e-02 1.281218e-01 1.597697e-01 2.220647e-01
## theta[587] 2.387323e-01 2.566703e-01 2.746634e-01 3.135383e-01
## theta[588] 9.440034e-02 1.219809e-01 1.528403e-01 2.387715e-01
## theta[589] 2.013170e-01 2.308952e-01 2.597600e-01 3.320652e-01
## theta[590] 9.148700e-02 1.137487e-01 1.379907e-01 1.914698e-01
## theta[591] 2.859923e-01 2.979498e-01 3.093665e-01 3.305250e-01
## theta[592] 6.322799e-02 7.964650e-02 9.789304e-02 1.422956e-01
## theta[593] 2.023911e-01 2.238567e-01 2.445396e-01 2.857222e-01
## theta[594] 2.438214e-01 2.561481e-01 2.701591e-01 2.940651e-01
## theta[595] 2.385203e-01 2.554201e-01 2.743293e-01 3.117118e-01
## theta[596] 2.147912e-01 2.264904e-01 2.391346e-01 2.640962e-01
## theta[597] 1.174717e-01 1.461432e-01 1.761316e-01 2.443106e-01
## theta[598] 2.383316e-01 2.515107e-01 2.634559e-01 2.839373e-01
## theta[599] 2.528727e-01 2.647311e-01 2.771090e-01 2.981504e-01
## theta[600] 2.121361e-01 2.305979e-01 2.506430e-01 2.858078e-01
## theta[601] 2.426831e-01 2.573372e-01 2.761480e-01 3.115168e-01
## theta[602] 2.823021e-01 2.937326e-01 3.046900e-01 3.276591e-01
## theta[603] 2.208210e-01 2.371309e-01 2.546010e-01 2.880735e-01
## theta[604] 2.257873e-01 2.492398e-01 2.746835e-01 3.212632e-01
## theta[605] 1.340305e-01 1.582137e-01 1.849504e-01 2.390764e-01
## theta[606] 2.314775e-01 2.514717e-01 2.728478e-01 3.188683e-01
## theta[607] 8.702843e-02 1.180689e-01 1.496756e-01 2.271223e-01
## theta[608] 1.337099e-01 1.557950e-01 1.787652e-01 2.306494e-01
## theta[609] 2.666431e-01 2.821519e-01 2.982919e-01 3.293319e-01
## theta[610] 2.167556e-01 2.399236e-01 2.613788e-01 3.019098e-01
## theta[611] 2.147475e-01 2.355890e-01 2.555932e-01 2.970659e-01
## theta[612] 2.496393e-01 2.711379e-01 2.920978e-01 3.319739e-01
## theta[613] 2.012831e-01 2.183677e-01 2.360157e-01 2.698300e-01
## theta[614] 8.138092e-02 1.029574e-01 1.270852e-01 1.767692e-01
## theta[615] 2.211706e-01 2.413972e-01 2.624553e-01 3.071064e-01
## theta[616] 8.696406e-02 1.132356e-01 1.432822e-01 2.245722e-01
## theta[617] 1.493801e-01 1.778779e-01 2.136034e-01 2.925828e-01
## theta[618] 2.779408e-01 2.885815e-01 2.998088e-01 3.229606e-01
## theta[619] 2.789348e-01 2.936902e-01 3.076719e-01 3.348563e-01
## theta[620] 2.243471e-01 2.470023e-01 2.702246e-01 3.161405e-01
## theta[621] 2.133867e-01 2.351410e-01 2.583865e-01 2.965244e-01
## theta[622] 8.727987e-02 1.147439e-01 1.485768e-01 2.238462e-01
## theta[623] 1.814856e-01 2.015731e-01 2.213888e-01 2.585704e-01
## theta[624] 2.358012e-01 2.470114e-01 2.586427e-01 2.820836e-01
## theta[625] 2.310745e-01 2.437754e-01 2.572417e-01 2.819365e-01
## theta[626] 2.424422e-01 2.623675e-01 2.826711e-01 3.250833e-01
## theta[627] 9.076504e-02 1.179933e-01 1.502509e-01 2.249141e-01
## theta[628] 3.146360e-01 3.270884e-01 3.400172e-01 3.661600e-01
## theta[629] 2.259049e-01 2.583719e-01 2.899650e-01 3.559143e-01
## theta[630] 8.894502e-02 1.191678e-01 1.528627e-01 2.235044e-01
## theta[631] 2.606488e-01 2.717599e-01 2.832141e-01 3.054820e-01
## theta[632] 9.278216e-02 1.206353e-01 1.525873e-01 2.298296e-01
## theta[633] 2.707726e-01 2.830179e-01 2.952793e-01 3.191203e-01
## theta[634] 2.467274e-01 2.595443e-01 2.732693e-01 3.018912e-01
## theta[635] 1.904878e-01 2.082289e-01 2.264502e-01 2.611152e-01
## theta[636] 2.186806e-01 2.345179e-01 2.514178e-01 2.826283e-01
## theta[637] 2.366254e-01 2.487867e-01 2.607314e-01 2.816206e-01
## theta[638] 2.615631e-01 2.725066e-01 2.852738e-01 3.063562e-01
## theta[639] 8.534761e-02 1.127677e-01 1.414446e-01 2.083474e-01
## theta[640] 3.082061e-01 3.197507e-01 3.319516e-01 3.542704e-01
## theta[641] 1.970392e-01 2.133725e-01 2.307213e-01 2.712193e-01
## theta[642] 2.654154e-01 2.776745e-01 2.904707e-01 3.157372e-01
## theta[643] 2.122706e-01 2.256306e-01 2.397291e-01 2.663575e-01
## theta[644] 2.380916e-01 2.498659e-01 2.606595e-01 2.869697e-01
## theta[645] 2.442834e-01 2.605826e-01 2.788965e-01 3.153073e-01
## theta[646] 2.105229e-01 2.277063e-01 2.455123e-01 2.782846e-01
## theta[647] 1.967209e-01 2.196505e-01 2.437369e-01 2.936105e-01
## theta[648] 9.279270e-02 1.197179e-01 1.517375e-01 2.215467e-01
## theta[649] 8.818266e-02 1.111212e-01 1.380770e-01 1.961534e-01
## theta[650] 2.428375e-01 2.647422e-01 2.880091e-01 3.312834e-01
## theta[651] 1.894425e-01 2.160660e-01 2.426986e-01 3.044694e-01
## theta[652] 9.166695e-02 1.187123e-01 1.511329e-01 2.339379e-01
## theta[653] 2.004229e-01 2.212179e-01 2.447166e-01 2.973179e-01
## theta[654] 7.738444e-02 9.474381e-02 1.165951e-01 1.651434e-01
## theta[655] 2.690835e-01 2.824567e-01 2.950009e-01 3.209747e-01
## theta[656] 2.322744e-01 2.435970e-01 2.549001e-01 2.759023e-01
## theta[657] 2.160046e-01 2.299141e-01 2.439996e-01 2.704019e-01
## theta[658] 1.820575e-01 2.022294e-01 2.216728e-01 2.616469e-01
## theta[659] 2.019887e-01 2.246927e-01 2.505624e-01 2.990039e-01
## theta[660] 1.012766e-01 1.306023e-01 1.623754e-01 2.362016e-01
## theta[661] 3.034486e-01 3.159432e-01 3.304130e-01 3.542227e-01
## theta[662] 9.101912e-02 1.175748e-01 1.469277e-01 2.160558e-01
## theta[663] 1.994030e-01 2.218499e-01 2.450688e-01 2.930373e-01
## theta[664] 2.555250e-01 2.698694e-01 2.858056e-01 3.135545e-01
## theta[665] 2.216479e-01 2.394425e-01 2.567605e-01 2.949449e-01
## theta[666] 1.955919e-01 2.159255e-01 2.347502e-01 2.731402e-01
## theta[667] 2.275996e-01 2.414764e-01 2.568344e-01 2.836335e-01
## theta[668] 2.403325e-01 2.572948e-01 2.737211e-01 3.040281e-01
## theta[669] 2.068344e-01 2.340206e-01 2.613752e-01 3.159375e-01
## theta[670] 9.409871e-02 1.218851e-01 1.571210e-01 2.369736e-01
## theta[671] 8.989523e-02 1.123797e-01 1.372199e-01 1.941815e-01
## theta[672] 9.084065e-02 1.203204e-01 1.552696e-01 2.349784e-01
## theta[673] 2.611257e-01 2.736002e-01 2.866127e-01 3.096043e-01
## theta[674] 9.327648e-02 1.207223e-01 1.516708e-01 2.298847e-01
## theta[675] 2.107714e-01 2.353841e-01 2.625442e-01 3.170406e-01
## theta[676] 1.651330e-01 1.809152e-01 1.958959e-01 2.234738e-01
## theta[677] 2.131848e-01 2.315298e-01 2.534786e-01 2.984415e-01
## theta[678] 2.763176e-01 2.900223e-01 3.032853e-01 3.292260e-01
## theta[679] 2.141793e-01 2.326323e-01 2.518977e-01 2.884154e-01
## theta[680] 2.577523e-01 2.684772e-01 2.803202e-01 3.024831e-01
## theta[681] 8.981979e-02 1.185671e-01 1.508176e-01 2.251155e-01
## theta[682] 2.495215e-01 2.714927e-01 2.941099e-01 3.411434e-01
## theta[683] 2.277706e-01 2.432165e-01 2.577406e-01 2.852363e-01
## theta[684] 2.593489e-01 2.707300e-01 2.822219e-01 3.026684e-01
## theta[685] 2.281968e-01 2.446083e-01 2.627622e-01 2.980963e-01
## theta[686] 2.222916e-01 2.399760e-01 2.583439e-01 2.956858e-01
## theta[687] 1.973697e-01 2.278710e-01 2.578486e-01 3.184097e-01
## theta[688] 2.020150e-01 2.156677e-01 2.304014e-01 2.575004e-01
## theta[689] 2.642008e-01 2.792904e-01 2.935900e-01 3.204719e-01
## theta[690] 9.595728e-02 1.167980e-01 1.440663e-01 1.958252e-01
## theta[691] 2.678504e-01 2.793650e-01 2.914471e-01 3.166134e-01
## theta[692] 1.888677e-01 2.068924e-01 2.250300e-01 2.588614e-01
## theta[693] 2.778484e-01 2.903353e-01 3.029289e-01 3.270145e-01
## theta[694] 7.555676e-02 9.344151e-02 1.120195e-01 1.503968e-01
## theta[695] 2.353647e-01 2.462691e-01 2.573930e-01 2.821141e-01
## theta[696] 2.658487e-01 2.814168e-01 2.966551e-01 3.303831e-01
## theta[697] 2.100167e-01 2.289213e-01 2.477922e-01 2.842561e-01
## theta[698] 2.565432e-01 2.821133e-01 3.116897e-01 3.645260e-01
## theta[699] 2.465713e-01 2.695970e-01 2.948869e-01 3.426431e-01
## theta[700] 2.151717e-01 2.321766e-01 2.477894e-01 2.842833e-01
## theta[701] 9.207575e-02 1.194978e-01 1.527573e-01 2.272210e-01
## theta[702] 1.010712e-01 1.299682e-01 1.639369e-01 2.425379e-01
## theta[703] 2.404577e-01 2.555530e-01 2.718247e-01 2.994551e-01
## theta[704] 2.952051e-01 3.059854e-01 3.181294e-01 3.421326e-01
## theta[705] 2.399247e-01 2.533881e-01 2.687794e-01 2.979810e-01
## theta[706] 2.488349e-01 2.614006e-01 2.738959e-01 3.023262e-01
## theta[707] 2.312171e-01 2.613984e-01 2.909586e-01 3.527245e-01
## theta[708] 2.426050e-01 2.551640e-01 2.678737e-01 2.906512e-01
## theta[709] 2.178862e-01 2.466558e-01 2.765245e-01 3.452257e-01
## theta[710] 2.031629e-01 2.194221e-01 2.361938e-01 2.676200e-01
## theta[711] 1.353216e-01 1.622829e-01 1.957772e-01 2.716483e-01
## theta[712] 1.036886e-01 1.337242e-01 1.659249e-01 2.358673e-01
## theta[713] 1.124798e-01 1.338206e-01 1.587262e-01 2.216351e-01
## theta[714] 2.285277e-01 2.421937e-01 2.549693e-01 2.809822e-01
## theta[715] 2.225432e-01 2.455869e-01 2.695894e-01 3.216173e-01
## theta[716] 2.628159e-01 2.773354e-01 2.919327e-01 3.281184e-01
## theta[717] 1.872882e-01 2.177073e-01 2.465940e-01 3.025512e-01
## theta[718] 9.097973e-02 1.221926e-01 1.557625e-01 2.286308e-01
## theta[719] 9.325864e-02 1.215256e-01 1.548337e-01 2.193448e-01
## theta[720] 2.211597e-01 2.381799e-01 2.558129e-01 2.913541e-01
## theta[721] 8.937476e-02 1.166010e-01 1.509311e-01 2.347781e-01
## theta[722] 2.233631e-01 2.337278e-01 2.453498e-01 2.683243e-01
## theta[723] 1.226066e-01 1.439466e-01 1.678501e-01 2.171343e-01
## theta[724] 1.037164e-01 1.312348e-01 1.653777e-01 2.430222e-01
## theta[725] 2.404726e-01 2.629034e-01 2.853375e-01 3.262121e-01
## theta[726] 2.168492e-01 2.428339e-01 2.694307e-01 3.173471e-01
## theta[727] 9.301554e-02 1.232069e-01 1.583871e-01 2.339412e-01
## theta[728] 9.267700e-02 1.198575e-01 1.538607e-01 2.233360e-01
## theta[729] 2.062013e-01 2.190581e-01 2.320054e-01 2.556350e-01
## theta[730] 2.917551e-01 3.040519e-01 3.163533e-01 3.369812e-01
## theta[731] 1.998923e-01 2.138744e-01 2.263448e-01 2.569162e-01
## theta[732] 2.606490e-01 2.760548e-01 2.917939e-01 3.282031e-01
## theta[733] 9.242812e-02 1.201281e-01 1.574446e-01 2.316099e-01
## theta[734] 2.374589e-01 2.547217e-01 2.739500e-01 3.090371e-01
## theta[735] 2.223684e-01 2.454945e-01 2.686867e-01 3.192018e-01
## theta[736] 6.838814e-02 8.720928e-02 1.069006e-01 1.505172e-01
## theta[737] 8.547687e-02 1.104776e-01 1.397270e-01 2.112044e-01
## theta[738] 1.249381e-01 1.467183e-01 1.698333e-01 2.193308e-01
## theta[739] 2.710913e-01 2.825604e-01 2.936427e-01 3.177726e-01
## theta[740] 2.364338e-01 2.580158e-01 2.811434e-01 3.221962e-01
## theta[741] 2.029691e-01 2.157548e-01 2.278954e-01 2.502435e-01
## theta[742] 8.934871e-02 1.114740e-01 1.368278e-01 1.847874e-01
## theta[743] 2.556609e-01 2.684647e-01 2.826244e-01 3.046761e-01
## theta[744] 2.115901e-01 2.288074e-01 2.460151e-01 2.850611e-01
## theta[745] 2.509785e-01 2.658427e-01 2.808112e-01 3.138297e-01
## theta[746] 2.159430e-01 2.296698e-01 2.449665e-01 2.742139e-01
## theta[747] 2.237496e-01 2.439028e-01 2.662176e-01 3.105747e-01
## theta[748] 2.163238e-01 2.447794e-01 2.760194e-01 3.280110e-01
## theta[749] 1.686312e-01 1.857997e-01 2.037152e-01 2.417506e-01
## theta[750] 2.567558e-01 2.697983e-01 2.827290e-01 3.079614e-01
## theta[751] 9.688706e-02 1.228611e-01 1.518128e-01 2.292798e-01
## theta[752] 1.801610e-01 1.960207e-01 2.125575e-01 2.460436e-01
## theta[753] 2.255536e-01 2.378477e-01 2.506115e-01 2.766274e-01
## theta[754] 2.438591e-01 2.592814e-01 2.741760e-01 3.062753e-01
## theta[755] 2.500982e-01 2.645884e-01 2.778267e-01 3.049956e-01
## theta[756] 8.737900e-02 1.085822e-01 1.314868e-01 1.754662e-01
## theta[757] 9.522423e-02 1.233942e-01 1.553210e-01 2.343663e-01
## theta[758] 2.251714e-01 2.453852e-01 2.672596e-01 3.108122e-01
## theta[759] 2.665582e-01 2.788855e-01 2.904014e-01 3.131738e-01
## theta[760] 2.706417e-01 2.867486e-01 3.019785e-01 3.332243e-01
## theta[761] 2.342758e-01 2.560146e-01 2.769453e-01 3.202284e-01
## theta[762] 9.206629e-02 1.208336e-01 1.554800e-01 2.361529e-01
## theta[763] 2.199760e-01 2.492013e-01 2.767872e-01 3.333394e-01
## theta[764] 1.090269e-01 1.368902e-01 1.722132e-01 2.476451e-01
## theta[765] 1.904767e-01 2.111267e-01 2.323627e-01 2.700564e-01
## theta[766] 9.656746e-02 1.258873e-01 1.575303e-01 2.228964e-01
## theta[767] 1.015987e-01 1.313390e-01 1.661716e-01 2.350038e-01
## theta[768] 2.452471e-01 2.589820e-01 2.730943e-01 3.013604e-01
## theta[769] 9.229199e-02 1.202532e-01 1.514784e-01 2.233629e-01
## theta[770] 2.401537e-01 2.565274e-01 2.725026e-01 3.078960e-01
## theta[771] 2.631247e-01 2.800018e-01 2.997379e-01 3.338148e-01
## theta[772] 2.335798e-01 2.486058e-01 2.638746e-01 2.931417e-01
## theta[773] 2.037793e-01 2.255408e-01 2.499142e-01 2.943972e-01
## theta[774] 2.025494e-01 2.292185e-01 2.576023e-01 3.156346e-01
## theta[775] 2.078304e-01 2.224467e-01 2.370724e-01 2.664595e-01
## theta[776] 2.284604e-01 2.407983e-01 2.546850e-01 2.806496e-01
## theta[777] 9.399027e-02 1.228129e-01 1.549873e-01 2.336643e-01
## theta[778] 2.345325e-01 2.519491e-01 2.710142e-01 3.065049e-01
## theta[779] 2.437250e-01 2.552984e-01 2.678967e-01 2.889232e-01
## theta[780] 9.541413e-02 1.151528e-01 1.375816e-01 1.949139e-01
## theta[781] 1.212509e-01 1.544961e-01 1.943415e-01 2.825416e-01
## theta[782] 2.020370e-01 2.161348e-01 2.307215e-01 2.623949e-01
## theta[783] 2.525008e-01 2.674819e-01 2.828086e-01 3.162296e-01
## theta[784] 2.680441e-01 2.788907e-01 2.896216e-01 3.135615e-01
## theta[785] 2.347929e-01 2.507473e-01 2.693225e-01 3.049294e-01
## theta[786] 2.159511e-01 2.290877e-01 2.432802e-01 2.726822e-01
## theta[787] 1.158349e-01 1.449568e-01 1.751947e-01 2.499789e-01
## theta[788] 1.692683e-01 1.976196e-01 2.262790e-01 2.927442e-01
## theta[789] 1.583911e-01 1.862028e-01 2.125222e-01 2.689375e-01
## theta[790] 9.200123e-02 1.211480e-01 1.547190e-01 2.160551e-01
## theta[791] 1.954462e-01 2.113092e-01 2.293942e-01 2.636614e-01
## theta[792] 9.193610e-02 1.197939e-01 1.526357e-01 2.255972e-01
## theta[793] 2.267546e-01 2.428141e-01 2.597442e-01 2.957256e-01
## theta[794] 2.176887e-01 2.452171e-01 2.726758e-01 3.282718e-01
## theta[795] 2.045774e-01 2.335830e-01 2.639079e-01 3.231482e-01
## theta[796] 1.986706e-01 2.191803e-01 2.422129e-01 2.896345e-01
## theta[797] 1.987152e-01 2.166349e-01 2.358278e-01 2.739548e-01
## theta[798] 1.848476e-01 2.058018e-01 2.293121e-01 2.700964e-01
## theta[799] 8.545493e-02 1.103151e-01 1.367884e-01 1.991147e-01
## theta[800] 2.144178e-01 2.420716e-01 2.701069e-01 3.305035e-01
## theta[801] 2.153346e-01 2.434673e-01 2.710915e-01 3.278168e-01
## theta[802] 9.044184e-02 1.194145e-01 1.522894e-01 2.307370e-01
## theta[803] 2.127175e-01 2.420433e-01 2.715880e-01 3.289281e-01
## theta[804] 1.462753e-01 1.710773e-01 1.997334e-01 2.518294e-01
## theta[805] 8.671572e-02 1.058543e-01 1.279165e-01 1.763783e-01
## theta[806] 9.511657e-02 1.213625e-01 1.514075e-01 2.348243e-01
## theta[807] 2.562229e-01 2.697644e-01 2.829792e-01 3.066892e-01
## theta[808] 2.251155e-01 2.412251e-01 2.565788e-01 2.903473e-01
## theta[809] 9.391195e-02 1.231673e-01 1.591496e-01 2.388206e-01
## theta[810] 1.168214e-01 1.446732e-01 1.810449e-01 2.541581e-01
## theta[811] 9.713859e-02 1.247207e-01 1.543759e-01 2.312323e-01
## theta[812] 5.620289e-02 7.282079e-02 9.053678e-02 1.374315e-01
## theta[813] 9.398644e-02 1.222916e-01 1.579220e-01 2.295445e-01
## theta[814] 1.034128e-01 1.329172e-01 1.642987e-01 2.395164e-01
## theta[815] 2.314273e-01 2.525158e-01 2.748921e-01 3.244302e-01
## theta[816] 2.412877e-01 2.605165e-01 2.798849e-01 3.175587e-01
## theta[817] 9.049556e-02 1.212962e-01 1.580479e-01 2.254489e-01
## theta[818] 2.267989e-01 2.468465e-01 2.675034e-01 3.065853e-01
## theta[819] 9.404690e-02 1.232473e-01 1.553093e-01 2.382615e-01
## theta[820] 2.917171e-01 3.045517e-01 3.166861e-01 3.378664e-01
## theta[821] 8.983844e-02 1.170483e-01 1.512441e-01 2.232757e-01
## theta[822] 2.160249e-01 2.365011e-01 2.588692e-01 3.017071e-01
## theta[823] 2.365043e-01 2.545324e-01 2.729397e-01 3.063310e-01
## theta[824] 2.375069e-01 2.552173e-01 2.725005e-01 3.089984e-01
## theta[825] 1.322266e-01 1.543994e-01 1.785878e-01 2.332246e-01
## theta[826] 2.162523e-01 2.330802e-01 2.505322e-01 2.838914e-01
## theta[827] 9.027424e-02 1.144734e-01 1.425989e-01 2.121084e-01
## theta[828] 1.013564e-01 1.217773e-01 1.434135e-01 1.867637e-01
## theta[829] 2.251939e-01 2.379922e-01 2.499426e-01 2.743861e-01
## theta[830] 2.532069e-01 2.722485e-01 2.919831e-01 3.318874e-01
## theta[831] 1.911660e-01 2.176646e-01 2.441211e-01 2.972840e-01
## theta[832] 2.261435e-01 2.396619e-01 2.520916e-01 2.803050e-01
## theta[833] 1.255032e-01 1.522640e-01 1.842318e-01 2.512325e-01
## theta[834] 8.969191e-02 1.169385e-01 1.482100e-01 2.146204e-01
## theta[835] 2.701792e-01 2.829219e-01 2.964379e-01 3.219806e-01
## theta[836] 2.063646e-01 2.237055e-01 2.426155e-01 2.822809e-01
## theta[837] 2.186969e-01 2.506914e-01 2.817274e-01 3.482120e-01
## theta[838] 2.199209e-01 2.339982e-01 2.482841e-01 2.768970e-01
## theta[839] 2.377359e-01 2.564386e-01 2.752815e-01 3.127481e-01
## theta[840] 2.242774e-01 2.422773e-01 2.609243e-01 2.985040e-01
## theta[841] 8.395119e-02 1.085231e-01 1.375326e-01 1.989006e-01
## theta[842] 1.004054e-01 1.305406e-01 1.652401e-01 2.491116e-01
## theta[843] 9.445424e-02 1.196969e-01 1.548461e-01 2.251029e-01
## theta[844] 8.405138e-02 1.102966e-01 1.407581e-01 2.075389e-01
## theta[845] 8.694849e-02 1.077630e-01 1.298157e-01 1.814093e-01
## theta[846] 2.217651e-01 2.372248e-01 2.536046e-01 2.837465e-01
## theta[847] 9.596095e-02 1.237287e-01 1.584787e-01 2.294139e-01
## theta[848] 1.216040e-01 1.549161e-01 1.891578e-01 2.699319e-01
## theta[849] 1.922063e-01 2.127781e-01 2.354443e-01 2.734884e-01
## theta[850] 1.062720e-01 1.393611e-01 1.751695e-01 2.612572e-01
## theta[851] 8.810012e-02 1.150973e-01 1.467729e-01 2.211135e-01
## theta[852] 1.289661e-01 1.511663e-01 1.750280e-01 2.191946e-01
## theta[853] 7.403515e-02 9.223202e-02 1.128306e-01 1.538026e-01
## theta[854] 2.379768e-01 2.550902e-01 2.737666e-01 3.143472e-01
## theta[855] 9.178189e-02 1.187432e-01 1.498305e-01 2.266269e-01
## theta[856] 2.427000e-01 2.658059e-01 2.901619e-01 3.377074e-01
## theta[857] 2.504337e-01 2.639819e-01 2.784531e-01 3.033929e-01
## theta[858] 2.646761e-01 2.788977e-01 2.945069e-01 3.264703e-01
## theta[859] 2.156538e-01 2.344260e-01 2.527411e-01 2.932654e-01
## theta[860] 2.487055e-01 2.619285e-01 2.755785e-01 3.002394e-01
## theta[861] 2.768998e-01 2.907425e-01 3.044459e-01 3.307333e-01
## theta[862] 1.852191e-01 2.083815e-01 2.316066e-01 2.767608e-01
## theta[863] 2.461075e-01 2.594906e-01 2.730289e-01 3.012497e-01
## theta[864] 2.330468e-01 2.534265e-01 2.767847e-01 3.246867e-01
## theta[865] 2.003309e-01 2.164029e-01 2.331517e-01 2.683325e-01
## theta[866] 9.757551e-02 1.223751e-01 1.517637e-01 2.195494e-01
## theta[867] 2.139735e-01 2.353238e-01 2.572967e-01 2.944486e-01
## theta[868] 9.027298e-02 1.221689e-01 1.573554e-01 2.525083e-01
## theta[869] 1.922604e-01 2.092661e-01 2.269602e-01 2.586565e-01
## theta[870] 2.555193e-01 2.775232e-01 2.989410e-01 3.416248e-01
## theta[871] 2.910446e-01 3.034421e-01 3.158313e-01 3.396064e-01
## theta[872] 1.990173e-01 2.210544e-01 2.439322e-01 2.910816e-01
## theta[873] 2.093807e-01 2.284692e-01 2.491221e-01 2.905154e-01
## theta[874] 2.612618e-01 2.801050e-01 2.987470e-01 3.356537e-01
## theta[875] 2.719134e-01 2.841554e-01 2.969654e-01 3.198362e-01
## theta[876] 2.589168e-01 2.702646e-01 2.818813e-01 3.035390e-01
## theta[877] 2.139600e-01 2.297796e-01 2.457165e-01 2.815250e-01
## theta[878] 1.242203e-01 1.459958e-01 1.700181e-01 2.187996e-01
## theta[879] 1.083215e-01 1.370589e-01 1.668562e-01 2.532335e-01
## theta[880] 2.412575e-01 2.521436e-01 2.632240e-01 2.867316e-01
## theta[881] 2.192372e-01 2.350741e-01 2.501754e-01 2.825940e-01
## theta[882] 9.706038e-02 1.282454e-01 1.613195e-01 2.466420e-01
## theta[883] 8.954104e-02 1.188740e-01 1.505382e-01 2.266476e-01
## theta[884] 2.094272e-01 2.346603e-01 2.634124e-01 3.201012e-01
## theta[885] 2.335652e-01 2.459502e-01 2.583942e-01 2.823969e-01
## theta[886] 1.306015e-01 1.560730e-01 1.881181e-01 2.577647e-01
## gamma[1] -1.116217e+00 -1.092166e+00 -1.068037e+00 -1.023600e+00
## gamma[2] -1.122286e+00 -1.094753e+00 -1.068941e+00 -1.016003e+00
## gamma[3] -1.091680e+00 -1.068998e+00 -1.046223e+00 -1.003571e+00
## gamma[4] -1.195250e+00 -1.168942e+00 -1.144002e+00 -1.097949e+00
## gamma[5] -1.075370e+00 -1.049713e+00 -1.024780e+00 -9.709247e-01
## gamma[6] -1.126767e+00 -1.103972e+00 -1.083695e+00 -1.042206e+00
## gamma[7] -1.947378e+00 -1.910580e+00 -1.869446e+00 -1.796998e+00
## gamma[8] -1.077111e+00 -1.054510e+00 -1.029831e+00 -9.782488e-01
## gamma[9] -1.130397e+00 -1.102554e+00 -1.073996e+00 -1.024073e+00
## alpha_w 8.724412e-03 2.017400e-02 3.184715e-02 5.248465e-02
## alpha_h -2.002119e-02 -1.014698e-02 2.271136e-03 2.194205e-02
## kappa[1] 9.235578e+01 1.057593e+02 1.218540e+02 1.612551e+02
## kappa[2] 8.856681e+01 1.038930e+02 1.201477e+02 1.589257e+02
## kappa[3] 8.082473e+01 9.392196e+01 1.089253e+02 1.457635e+02
## kappa[4] 7.716422e+01 8.866078e+01 1.023437e+02 1.297261e+02
## kappa[5] 7.784985e+01 9.195951e+01 1.075265e+02 1.421060e+02
## kappa[6] 9.151975e+01 1.047872e+02 1.203633e+02 1.568054e+02
## kappa[7] 4.291129e+01 5.100324e+01 5.996130e+01 8.259997e+01
## kappa[8] 9.136948e+01 1.085185e+02 1.284875e+02 1.649810e+02
## kappa[9] 7.415940e+01 8.673494e+01 1.013781e+02 1.368378e+02
## lp__ -8.743234e+04 -8.740717e+04 -8.737957e+04 -8.733013e+04
## n_eff Rhat
## theta[1] 2574.2233 0.9996139
## theta[2] 2575.3408 1.0004334
## theta[3] 3398.8125 0.9984184
## theta[4] 2647.5748 0.9998957
## theta[5] 2930.0942 0.9987725
## theta[6] 3653.2360 1.0008400
## theta[7] 2453.6873 1.0001072
## theta[8] 3114.5259 0.9980857
## theta[9] 3424.3880 0.9995881
## theta[10] 2647.2132 0.9998986
## theta[11] 2295.0168 0.9982430
## theta[12] 3251.8448 0.9991356
## theta[13] 2693.8937 0.9993625
## theta[14] 2582.4450 0.9982356
## theta[15] 2833.4626 0.9984091
## theta[16] 2728.3969 0.9982424
## theta[17] 2899.2031 0.9989393
## theta[18] 3952.3844 0.9980782
## theta[19] 2438.7856 0.9997186
## theta[20] 3606.7835 0.9983621
## theta[21] 2850.2377 0.9994896
## theta[22] 2580.5389 1.0005145
## theta[23] 3270.8392 1.0000979
## theta[24] 3633.7950 0.9987422
## theta[25] 3053.5624 0.9992465
## theta[26] 3224.5323 0.9980854
## theta[27] 3339.3615 0.9982612
## theta[28] 2288.6795 0.9994352
## theta[29] 2906.9648 0.9984924
## theta[30] 3608.1015 1.0005500
## theta[31] 3800.8162 0.9987966
## theta[32] 3125.2587 0.9982042
## theta[33] 3080.8500 0.9990144
## theta[34] 3992.3182 0.9981661
## theta[35] 2261.5902 0.9989677
## theta[36] 3313.8977 0.9984058
## theta[37] 2870.3817 0.9984195
## theta[38] 2186.5542 0.9993061
## theta[39] 2617.7113 0.9989342
## theta[40] 2237.7836 0.9989847
## theta[41] 3835.3148 0.9989732
## theta[42] 2501.5977 0.9983002
## theta[43] 2871.4120 0.9987626
## theta[44] 2418.3736 0.9997392
## theta[45] 3526.3518 0.9988862
## theta[46] 2378.6047 0.9987257
## theta[47] 3287.5988 0.9983484
## theta[48] 2754.0181 0.9989368
## theta[49] 4257.5889 0.9990439
## theta[50] 3473.9875 0.9987508
## theta[51] 3990.4043 0.9997316
## theta[52] 2615.7395 0.9993548
## theta[53] 2038.1596 0.9991528
## theta[54] 3480.9015 0.9988001
## theta[55] 2773.2006 0.9993563
## theta[56] 3275.6452 0.9981017
## theta[57] 2531.3620 0.9991099
## theta[58] 2917.3759 0.9983378
## theta[59] 2806.2356 0.9984676
## theta[60] 2924.3384 0.9990905
## theta[61] 2763.0618 0.9997999
## theta[62] 2809.5043 0.9988260
## theta[63] 3025.2700 0.9983251
## theta[64] 2963.2641 0.9984835
## theta[65] 4391.0905 0.9984350
## theta[66] 2809.8852 0.9991829
## theta[67] 3047.4771 0.9986460
## theta[68] 3481.5692 0.9986425
## theta[69] 2602.6428 0.9987186
## theta[70] 2150.6444 0.9985719
## theta[71] 3379.2644 0.9992305
## theta[72] 4224.2472 0.9983474
## theta[73] 2476.2184 0.9982991
## theta[74] 3845.1574 0.9990304
## theta[75] 3280.0394 0.9989146
## theta[76] 1828.4152 0.9985719
## theta[77] 3346.0222 0.9987536
## theta[78] 2920.0055 0.9984322
## theta[79] 2472.4876 0.9997645
## theta[80] 3005.3123 1.0000448
## theta[81] 2902.6275 0.9985226
## theta[82] 2377.0531 0.9989460
## theta[83] 3115.5566 0.9988652
## theta[84] 2995.7602 0.9987769
## theta[85] 2548.7437 0.9996302
## theta[86] 3462.6600 0.9987629
## theta[87] 2264.9333 0.9998644
## theta[88] 2590.3289 0.9986693
## theta[89] 2358.4398 1.0000695
## theta[90] 3035.5317 0.9984712
## theta[91] 3448.4015 0.9987331
## theta[92] 3755.8570 0.9992617
## theta[93] 2421.3140 0.9981588
## theta[94] 2930.4075 0.9995914
## theta[95] 2486.4639 0.9989734
## theta[96] 2433.9496 0.9986851
## theta[97] 3285.7389 0.9983304
## theta[98] 2622.1472 0.9995907
## theta[99] 2734.7938 0.9989684
## theta[100] 3709.7707 0.9987754
## theta[101] 3459.4955 0.9986876
## theta[102] 2837.8941 0.9989302
## theta[103] 3463.5000 0.9987740
## theta[104] 2586.7264 0.9987284
## theta[105] 3035.8484 0.9994899
## theta[106] 2461.1040 0.9989305
## theta[107] 1720.5521 0.9995127
## theta[108] 2761.7110 0.9985303
## theta[109] 2820.8758 0.9993743
## theta[110] 3589.8911 0.9984498
## theta[111] 2949.6312 0.9985306
## theta[112] 2718.6427 0.9987738
## theta[113] 3867.8810 0.9991139
## theta[114] 3013.2618 0.9994502
## theta[115] 2504.2871 0.9990952
## theta[116] 2475.4793 0.9988717
## theta[117] 2415.9451 0.9994192
## theta[118] 2745.2152 0.9988003
## theta[119] 2066.5554 1.0005935
## theta[120] 3366.8491 0.9992406
## theta[121] 2944.8425 0.9984381
## theta[122] 2652.1551 0.9995911
## theta[123] 3240.4763 0.9982303
## theta[124] 2560.2344 0.9991925
## theta[125] 2461.7691 0.9985046
## theta[126] 2716.6324 0.9986303
## theta[127] 3373.8093 0.9995870
## theta[128] 2582.2507 0.9984958
## theta[129] 2741.5195 0.9984282
## theta[130] 3067.6965 0.9982709
## theta[131] 2923.7223 1.0008275
## theta[132] 3462.9470 0.9983801
## theta[133] 3001.6050 0.9983795
## theta[134] 2339.1074 0.9995984
## theta[135] 2789.7465 0.9982994
## theta[136] 2583.0049 0.9991827
## theta[137] 3401.1881 0.9981720
## theta[138] 3624.6382 1.0002032
## theta[139] 2613.8113 0.9998961
## theta[140] 3671.6871 0.9982934
## theta[141] 3783.1210 0.9984646
## theta[142] 3044.4546 0.9994750
## theta[143] 4259.4157 0.9984748
## theta[144] 2681.9806 0.9992563
## theta[145] 4388.5759 0.9984940
## theta[146] 3146.0532 0.9983345
## theta[147] 3033.3000 0.9992491
## theta[148] 3484.4387 0.9987186
## theta[149] 2977.8561 0.9990087
## theta[150] 2525.0250 0.9981805
## theta[151] 2733.0816 0.9990126
## theta[152] 2806.9239 0.9987227
## theta[153] 3280.2803 0.9985578
## theta[154] 2449.1807 0.9991424
## theta[155] 2840.2671 0.9989848
## theta[156] 3048.0039 0.9992258
## theta[157] 2876.3916 0.9995140
## theta[158] 4277.1406 0.9988278
## theta[159] 2218.0574 0.9985903
## theta[160] 3171.1124 0.9993023
## theta[161] 2670.2772 0.9990120
## theta[162] 3138.8262 0.9999409
## theta[163] 1880.6911 1.0002153
## theta[164] 2876.2342 0.9993448
## theta[165] 2884.8285 0.9990949
## theta[166] 2739.0757 1.0005261
## theta[167] 2945.0055 0.9984059
## theta[168] 3001.2116 0.9986214
## theta[169] 3238.2339 0.9986580
## theta[170] 2301.0802 0.9991629
## theta[171] 2122.1226 0.9989149
## theta[172] 3387.1629 0.9996830
## theta[173] 2837.6666 0.9985067
## theta[174] 2647.1953 0.9999002
## theta[175] 2358.5209 0.9989498
## theta[176] 2992.1125 0.9981390
## theta[177] 3971.3768 0.9981060
## theta[178] 3030.5198 0.9993883
## theta[179] 3154.8791 0.9982580
## theta[180] 2118.0572 0.9994228
## theta[181] 3039.8097 0.9990216
## theta[182] 3000.8263 0.9983771
## theta[183] 2173.3703 0.9993368
## theta[184] 2228.4668 1.0000075
## theta[185] 2729.4301 0.9981411
## theta[186] 3064.3804 0.9995580
## theta[187] 2354.8637 0.9993558
## theta[188] 2541.5242 1.0005922
## theta[189] 2789.5633 0.9995776
## theta[190] 2514.4343 0.9997454
## theta[191] 2772.7744 0.9991136
## theta[192] 2688.1068 0.9989156
## theta[193] 3363.8700 0.9993436
## theta[194] 2847.8224 0.9993126
## theta[195] 1903.1994 0.9986911
## theta[196] 3805.1455 0.9991158
## theta[197] 1872.9152 0.9997089
## theta[198] 1994.0799 0.9993511
## theta[199] 2425.2121 0.9991511
## theta[200] 2780.6577 0.9990566
## theta[201] 3007.3040 0.9991892
## theta[202] 2639.9246 0.9984357
## theta[203] 3541.5557 0.9986176
## theta[204] 3416.9671 0.9988494
## theta[205] 3050.0002 0.9985734
## theta[206] 2790.6981 0.9981468
## theta[207] 3949.3513 0.9987229
## theta[208] 2984.2822 0.9984385
## theta[209] 3532.5411 1.0006520
## theta[210] 3504.8865 0.9985344
## theta[211] 3097.3117 0.9987818
## theta[212] 2614.5267 0.9987149
## theta[213] 3247.9902 0.9989566
## theta[214] 3361.5411 0.9999951
## theta[215] 2497.6397 0.9990675
## theta[216] 2507.1956 0.9997600
## theta[217] 3288.7715 0.9994783
## theta[218] 2883.3614 0.9994418
## theta[219] 2316.8899 1.0001997
## theta[220] 1833.0415 0.9989365
## theta[221] 3057.8694 1.0001124
## theta[222] 2543.9335 0.9999107
## theta[223] 3301.2856 0.9986021
## theta[224] 2406.1872 1.0004224
## theta[225] 1876.1017 1.0003857
## theta[226] 3259.3595 0.9990582
## theta[227] 2764.6643 0.9992292
## theta[228] 4153.7191 0.9982042
## theta[229] 2483.0883 0.9992663
## theta[230] 3306.1376 0.9990803
## theta[231] 2783.6194 0.9987832
## theta[232] 3376.8772 0.9985636
## theta[233] 2588.8044 0.9991228
## theta[234] 4767.9306 0.9991920
## theta[235] 2413.0546 0.9985675
## theta[236] 3191.8792 0.9983864
## theta[237] 3300.7148 0.9985336
## theta[238] 3197.7326 0.9990212
## theta[239] 3221.2563 0.9992220
## theta[240] 3667.4996 0.9981284
## theta[241] 2985.5769 0.9991590
## theta[242] 2532.2504 1.0004048
## theta[243] 3293.5592 0.9988983
## theta[244] 3499.6593 0.9981758
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## theta[728] 2509.5375 0.9983533
## theta[729] 2601.8920 0.9982411
## theta[730] 2773.4201 0.9985818
## theta[731] 3462.8507 0.9987762
## theta[732] 2774.2380 0.9984241
## theta[733] 2493.8134 0.9982820
## theta[734] 3118.3623 0.9986624
## theta[735] 2724.6514 0.9989514
## theta[736] 2116.5471 0.9992338
## theta[737] 2524.2164 0.9986842
## theta[738] 2754.6911 1.0000368
## theta[739] 2413.5645 0.9996186
## theta[740] 3501.6685 0.9986890
## theta[741] 5009.1635 0.9994686
## theta[742] 2405.0758 0.9989087
## theta[743] 3474.0647 0.9982610
## theta[744] 3296.8748 0.9987249
## theta[745] 3481.9753 0.9993931
## theta[746] 3548.7480 0.9986610
## theta[747] 2543.4582 0.9993728
## theta[748] 2302.2468 0.9992397
## theta[749] 2135.5403 1.0002040
## theta[750] 3638.1243 0.9986665
## theta[751] 2373.9082 0.9987235
## theta[752] 2643.5534 0.9987467
## theta[753] 3344.2101 0.9985107
## theta[754] 3102.4057 0.9985813
## theta[755] 3569.6579 0.9984669
## theta[756] 3296.0198 0.9986583
## theta[757] 2843.3647 0.9986551
## theta[758] 2613.2257 0.9985117
## theta[759] 2650.4659 1.0010400
## theta[760] 3240.6731 0.9986283
## theta[761] 2966.8655 0.9994656
## theta[762] 2911.8568 1.0006900
## theta[763] 2415.1969 0.9994206
## theta[764] 2742.7104 0.9987435
## theta[765] 3354.0808 1.0000874
## theta[766] 2362.9163 0.9984001
## theta[767] 3176.9639 0.9992143
## theta[768] 3081.3232 0.9992742
## theta[769] 2183.5346 1.0000912
## theta[770] 3875.2148 0.9986592
## theta[771] 2875.1306 0.9987687
## theta[772] 3069.6549 0.9989908
## theta[773] 3482.5419 0.9991018
## theta[774] 2233.2707 0.9984166
## theta[775] 4135.7580 0.9982530
## theta[776] 3347.0501 0.9987034
## theta[777] 2753.0526 0.9996168
## theta[778] 3433.1970 0.9988081
## theta[779] 2778.8631 0.9999581
## theta[780] 2894.2892 0.9984565
## theta[781] 2065.8530 0.9989333
## theta[782] 3685.2978 0.9988503
## theta[783] 3158.3331 0.9987644
## theta[784] 2768.5825 0.9992324
## theta[785] 3549.4337 0.9982869
## theta[786] 2566.1914 0.9995906
## theta[787] 2882.2501 0.9997650
## theta[788] 1953.5575 0.9987930
## theta[789] 2396.3892 0.9983949
## theta[790] 2552.9556 0.9985587
## theta[791] 3511.2694 1.0000964
## theta[792] 2144.3543 0.9992068
## theta[793] 2705.8024 0.9985497
## theta[794] 3488.5336 0.9993804
## theta[795] 3390.5830 0.9993335
## theta[796] 2736.9857 0.9988998
## theta[797] 3140.3882 0.9988638
## theta[798] 2755.0689 0.9997443
## theta[799] 2260.4730 0.9981942
## theta[800] 4180.1433 0.9988363
## theta[801] 2812.7909 0.9984984
## theta[802] 2159.1971 0.9989463
## theta[803] 3396.2253 0.9996908
## theta[804] 2754.7494 0.9988512
## theta[805] 3142.4524 0.9991637
## theta[806] 2523.7202 0.9985642
## theta[807] 3168.8606 0.9982415
## theta[808] 2881.8710 0.9996804
## theta[809] 2515.8640 0.9996006
## theta[810] 3081.4708 0.9991890
## theta[811] 2906.5017 0.9995718
## theta[812] 1257.6744 1.0003749
## theta[813] 2787.0309 0.9989665
## theta[814] 2565.9255 0.9982681
## theta[815] 2464.8576 1.0000594
## theta[816] 2718.8815 0.9985606
## theta[817] 2263.3369 0.9998203
## theta[818] 3007.4443 0.9983370
## theta[819] 2221.2056 0.9983906
## theta[820] 3551.7472 0.9981579
## theta[821] 2582.4462 0.9981638
## theta[822] 3121.2985 0.9981535
## theta[823] 3436.6755 0.9994454
## theta[824] 2848.1689 0.9999276
## theta[825] 3242.8641 0.9990805
## theta[826] 2178.4511 0.9981574
## theta[827] 2456.4197 0.9985996
## theta[828] 2878.8720 0.9990646
## theta[829] 3697.7524 0.9989365
## theta[830] 3114.4698 0.9990524
## theta[831] 2884.2340 0.9993835
## theta[832] 2829.8094 0.9992852
## theta[833] 3226.8014 0.9991941
## theta[834] 2948.8697 0.9998682
## theta[835] 3418.7459 0.9987565
## theta[836] 3988.9603 0.9982948
## theta[837] 2293.5108 0.9983709
## theta[838] 2943.7318 0.9995395
## theta[839] 2814.5700 1.0003883
## theta[840] 2537.2640 0.9996969
## theta[841] 1832.3763 1.0005318
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## theta[843] 2296.3939 0.9989930
## theta[844] 2846.4016 0.9982166
## theta[845] 2873.8932 0.9990261
## theta[846] 3275.2231 0.9984842
## theta[847] 2691.8451 0.9981664
## theta[848] 2568.6350 0.9997146
## theta[849] 3097.6940 0.9981818
## theta[850] 2282.5630 1.0030326
## theta[851] 2607.5936 0.9998541
## theta[852] 2946.8201 0.9983725
## theta[853] 2342.0167 0.9999764
## theta[854] 2941.2106 0.9986752
## theta[855] 2401.9311 0.9992501
## theta[856] 3287.8436 0.9990943
## theta[857] 2804.1718 0.9983572
## theta[858] 2747.7428 0.9989966
## theta[859] 3293.4623 0.9998422
## theta[860] 2542.1293 1.0009551
## theta[861] 3084.6281 0.9986769
## theta[862] 2186.7377 0.9981621
## theta[863] 3188.4933 0.9987540
## theta[864] 2913.5639 0.9985247
## theta[865] 3639.7598 0.9990696
## theta[866] 2728.8963 0.9984847
## theta[867] 3078.0212 0.9981837
## theta[868] 2070.7395 0.9983402
## theta[869] 3225.8852 0.9996031
## theta[870] 2535.4181 0.9995596
## theta[871] 2771.7000 0.9984106
## theta[872] 2796.4278 0.9991892
## theta[873] 2526.3506 0.9989060
## theta[874] 2850.1070 0.9989160
## theta[875] 3103.9994 0.9994415
## theta[876] 2978.0287 0.9984676
## theta[877] 2371.7950 0.9985327
## theta[878] 3238.4020 0.9991712
## theta[879] 2977.2402 0.9980384
## theta[880] 2835.0048 0.9985260
## theta[881] 2997.9497 0.9983325
## theta[882] 2857.2922 0.9984873
## theta[883] 1806.0522 0.9995217
## theta[884] 3887.5191 0.9993571
## theta[885] 4144.9669 0.9993741
## theta[886] 2587.2340 0.9999498
## gamma[1] 1866.0538 1.0025952
## gamma[2] 1631.7807 0.9997124
## gamma[3] 1897.0647 0.9989797
## gamma[4] 1234.3210 0.9981330
## gamma[5] 1830.7881 0.9991353
## gamma[6] 1727.7619 1.0008323
## gamma[7] 601.8236 1.0035979
## gamma[8] 2074.9191 0.9996453
## gamma[9] 1975.9032 0.9992053
## alpha_w 1199.2116 1.0007102
## alpha_h 1095.7301 0.9985817
## kappa[1] 1225.9617 1.0011755
## kappa[2] 1002.2103 0.9982192
## kappa[3] 1273.9688 0.9995029
## kappa[4] 800.8477 0.9990962
## kappa[5] 1401.4157 0.9989160
## kappa[6] 1037.5179 1.0010803
## kappa[7] 142.0912 1.0084091
## kappa[8] 1744.0529 1.0014872
## kappa[9] 1088.3032 1.0003990
## lp__ 160.0836 1.0101251Pitchers still don’t bat well
## ci_level: 0.8 (80% intervals)## outer_level: 0.95 (95% intervals)
Recall the mean of \(\theta_i\) is \[\begin{aligned} \mu_i = \frac{1}{1 + \exp(-\gamma_{p_i} - \alpha_h h_i - \alpha_w w_i)} . \end{aligned}\]
gamma_samples <- rstan::extract(logistic_fit)$gamma
colnames(gamma_samples) <- levels(batplus$PriPos)
mu_samples <- 1/(1+exp(-gamma_samples))
boxplot(mu_samples, ylab="posterior distrn, mean batting avg", las=3)
No effect of height or weight
## ci_level: 0.8 (80% intervals)## outer_level: 0.95 (95% intervals)
Same \(\theta\) as before?
new_post_theta <- colMeans(rstan::extract(logistic_fit)$theta)
old_post_theta <- batting$post_mean[match(batplus$Player, batting$Player)]
plot(old_post_theta, new_post_theta, xlab="theta, model 1", ylab="theta, model 2")
abline(0,1, col='red')
Conclusions
We don’t have good evidence that batting average varies substantially with height or weight.
Changing the model from a Beta prior on class means to a logistic transform of a linear model did not substantially affect the results.
Stochastic minute
If \(X \sim \Cauchy(\text{center}=\mu, \text{scale}=\sigma)\), then \(X\) has probability density \[\begin{aligned} f(x \given \mu, \sigma) = \frac{1}{\pi\left( 1 + \left( \frac{x - \mu}{\sigma} \right)^2 \right)} . \end{aligned}\]
The Cauchy is a good example of a distribution with “heavy tails”: rare, very large values.
If \(Z \sim \Normal(0, 1)\) and \(X ~ \Normal(0,1/Z)\) then \(X \sim \Cauchy(0,1)\).
If \(X_1, X_2, \ldots, X_n\) are independent \(\Cauchy(0,1)\) then \(\max(X_1, \ldots, X_n)\) is of size \(n\).
If \(X_1, X_2, \ldots, X_n\) are independent \(\Cauchy(0,1)\) then \[\begin{aligned} \frac{1}{n} \left(X_1 + \cdots + X_n\right) \sim \Cauchy(0,1) . \end{aligned}\]
- If \(X_1, X_2, \ldots, X_n\) are independent \(\Cauchy(0,1)\) then \[\begin{aligned} \frac{1}{n} \left(X_1 + \cdots + X_n\right) \sim \Cauchy(0,1) . \end{aligned}\]
Wait, what?!?
A single value has the same distribution as the mean of 1,000 of them?
Let’s look:
\(X \sim \Normal(0,1)\)
\(X \sim \Cauchy(0,1)\)
Another way to look at it: extreme values
n <- 100
plot(c(cummax(rcauchy(n))), type='l', ylab='max value so far', xlab='number of samples', col='red')
lines(c(cummax(rnorm(n))), col='black')
legend("bottomright", lty=1, col=c('black', 'red'), legend=c('normal', 'cauchy'))
Another way to look at it: extreme values
n <- 1000
plot(c(cummax(rcauchy(n))), type='l', ylab='max value so far', xlab='number of samples', col='red')
lines(c(cummax(rnorm(n))), col='black')
legend("bottomright", lty=1, col=c('black', 'red'), legend=c('normal', 'cauchy'))
Another way to look at it: extreme values
n <- 1e6
plot(c(cummax(rcauchy(n))), type='l', ylab='max value so far', xlab='number of samples', col='red')
lines(c(cummax(rnorm(n))), col='black')
legend("bottomright", lty=1, col=c('black', 'red'), legend=c('normal', 'cauchy'))
Exercise: mixture of error rates.
Suppose you are measuring relative metabolic rates of mice in the wild. Because life is complicated, the accuracy of your measurements varies widely. A model of the measured rate, \(R_i\), for a mouse at temperature \(T_i\) is \[\begin{aligned} R &\sim \Normal(120 + 0.7 * (T_i - 37), 1/E) \\ E &\sim \Normal(0, 1) . \end{aligned}\]
Simulate 200 measurements from this model, for temperatures between 36 and 38, and try to infer the true slope (0.7).
Robust regression
Standard linear regression
\[\begin{aligned} \hat y_i &= b_0 + b_1 x_i \\ y_i &\sim \Normal(\hat y_i, \sigma^2) . \end{aligned}\]
Simulate data:
Standard linear regression
## user system elapsed
## 0.004 0.000 0.003##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.61892 -0.38356 -0.02138 0.38431 1.20547
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.99293 0.03718 26.71 <2e-16 ***
## x 2.03029 0.03608 56.27 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5257 on 198 degrees of freedom
## Multiple R-squared: 0.9411, Adjusted R-squared: 0.9408
## F-statistic: 3166 on 1 and 198 DF, p-value: < 2.2e-16with Stan
slr_block <- "
data {
int N;
vector[N] x;
vector[N] y;
}
parameters {
real b0;
real b1;
real<lower=0> sigma;
}
model {
y ~ normal(b0 + b1*x, sigma);
}"
system.time(
stanlr <- stan(model_code=slr_block,
data=list(N=length(x), x=x, y=y), iter=1e3))## user system elapsed
## 65.016 3.418 83.269## Inference for Stan model: 3b05fdd2691a18718e9286fd7f1e1ac1.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## b0 0.99 0.00 0.04 0.92 0.97 0.99 1.02 1.06 2054 1
## b1 2.03 0.00 0.04 1.96 2.01 2.03 2.06 2.10 1981 1
## sigma 0.53 0.00 0.03 0.48 0.51 0.53 0.55 0.58 2060 1
## lp__ 27.49 0.04 1.22 24.26 26.93 27.82 28.36 28.85 1006 1
##
## Samples were drawn using NUTS(diag_e) at Thu Nov 21 09:45:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).Cauchy noise?
Relative axon growth for neurons after \(x\) hours:
Standard linear regression
## user system elapsed
## 0.002 0.000 0.001##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -65.973 -0.877 0.120 1.312 66.074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.8918 0.5740 1.554 0.122
## x 0.7230 0.6172 1.171 0.243
##
## Residual standard error: 8.117 on 198 degrees of freedom
## Multiple R-squared: 0.006883, Adjusted R-squared: 0.001868
## F-statistic: 1.372 on 1 and 198 DF, p-value: 0.2428with Stan
srr_block <- "
data {
int N;
vector[N] x;
vector[N] y;
}
parameters {
real b0;
real b1;
real<lower=0> sigma;
}
model {
y ~ cauchy(b0 + b1*x, sigma);
}"
system.time(
stanrr <- stan(model_code=srr_block,
data=list(N=length(x), x=x, y=y), iter=1e3))## user system elapsed
## 61.090 3.048 79.266## Inference for Stan model: cd6e07d98bffcf77ad2b924fcd0f1723.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff
## b0 1.02 0.00 0.06 0.92 0.98 1.02 1.06 1.14 1465
## b1 2.03 0.00 0.07 1.89 1.98 2.03 2.08 2.16 1824
## sigma 0.58 0.00 0.06 0.47 0.54 0.58 0.61 0.69 1935
## lp__ -173.78 0.04 1.20 -177.03 -174.34 -173.51 -172.88 -172.36 1116
## Rhat
## b0 1
## b1 1
## sigma 1
## lp__ 1
##
## Samples were drawn using NUTS(diag_e) at Thu Nov 21 09:47:07 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
Wrap-up
Modeling, and Stan
How well a statistical method works depends on the situation.
We can describe the “situation” with a probability model.
Inference usually works best if the probabilistic model reflects reality .
Stan lets you do inference using (almost) arbitrary models.
Explicit models make it easy to simulate, and therefore test your methods.
Hierarchical Bayesian models
It is often possible to infer things about populations that we can’t infer about individuals.
Doing so leads to sharing of information (or, “power”) between samples, and can improve accuracy.
Priors (and hyperpriors) on individual parameters provides a good way to do this.