Motivating problem: more coins

Suppose now we have data from \(n\) different coins from the same source. We don’t assume they have the same \(\theta\), but don’t know what its distribution is, so try to learn it.

\[\begin{aligned} Z_i &\sim \Binom(N_i, \theta_i) \\ \theta_i &\sim \Beta(\alpha, \beta) \\ \alpha &\sim \Unif(0, 100) \\ \beta &\sim \Unif(0, 100) \end{aligned}\]

Goal: find the posterior distribution of \(\alpha\), \(\beta\).

When you can’t do the integrals: MCMC

Goal: Given:

• a model with parameters \(\theta\),
• a prior distribution \(p(\theta)\) on \(\theta\), and
• data, \(D\),

“find”/ask questions of the posterior distribution on \(\theta\),

\[\begin{aligned} p(\theta \given D) = \frac{ p(D \given \theta) p(\theta) }{ p(D) } . \end{aligned}\]

Toy example:

(from beta-binomial coin example)

Do we think that \(\theta < 0.5\)?

(before:)

pbeta(0.5, post_a, post_b)

(now:)

mean(rbeta(1e6, post_a, post_b) < 0.5)

How? Markov chain Monte Carlo!

i.e., “random-walk-based stochastic integration”

Example: Gibbs sampling for uniform distribution on a region. (picture)

Overview of MCMC

Produces a random sequence of samples \(\theta_1, \theta_2, \ldots, \theta_N\).

0. Begin somewhere (at \(\theta_1\)).

At each step, starting at \(\theta_k\):

1. Propose a new location (nearby?): \(\theta_k'\)

2. Decide whether to accept it.

   • if so: set \(\theta_{k+1} \leftarrow \theta_k'\)
   • if not: set \(\theta_{k+1} \leftarrow \theta_k\)

3. Set \(k \leftarrow k+1\); if \(k=N\) then stop.

Key concepts

• Rules are chosen so that \(p(\theta \given D)\) is the stationary distribution (long-run average!) of the random walk (the “Markov chain”).

• The chain must mix fast enough so the distribution of visited states converges to \(p(\theta \given D)\).

• Because of autocorrelation, \((\theta_1, \theta_2, \ldots, \theta_N)\) are not \(N\) independent samples: they are roughly equivalent to \(N_\text{eff} < N\) independent samples.

• For better mixing, acceptance probabilities should not be too high or too low.

• Starting several chains far apart can help diagnose failure to mix: Gelman’s \(r\) quantifies how different they are.

On your feet

Three people, with randomness provided by others:

1. Pick a random \(\{N,S,E,W\}\).

2. Take a step in that direction,

   • unless you’d run into a wall or a table.

Now:

1. Pick a random \(\{N,S,E,W\}\).

2. Take a \(1+\Poisson(5)\) number of steps in that direction,

   • unless you’d run into a wall or a table.

How it works

Imagine the walkers are on a hill, and:

1. Pick a random \(\{N,S,E,W\}\).

2. If

   • the step is uphill, then take it.
   • the step is downhill, then flip a \(p\)-coin; if you get Heads, stay were you are.

What would this do?

“Quick and easy” MCMC: Stan

The skeletal Stan program

data {
    // stuff you input
}
transformed data {
    // stuff that's calculated from the data (just once, at the start)
}
parameters {
    // stuff you want to learn the posterior distribution of
}
transformed parameters {
    // stuff that's calculated from the parameters (at every step)
}
model {
    // the action!
}
generated quantities {
    // stuff you want computed also along the way
}

From before:

We’ve flipped a coin 10 times and got 6 Heads. We think the coin is close to fair, so put a \(\Beta(20,20)\) prior on it’s probability of heads, and want the posterior distribution.

\[\begin{aligned} Z &\sim \Binom(10, \theta) \\ \theta &\sim \Beta(20, 20) \end{aligned}\] Sample from \[\theta \given Z = 6\]

Running the MCMC: rstan

library(rstan)
fit <- stan(model_code=stan_block,  # stan block from above
            data=list(N=10, Z=6),
            chains=3, iter=10000)

lp__ is the log posterior density. Note n_eff.

print(fit)

## Inference for Stan model: 5f3115ac9593cded393e67d2d0e3ce84.
## 3 chains, each with iter=10000; warmup=5000; thin=1; 
## post-warmup draws per chain=5000, total post-warmup draws=15000.
## 
##         mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
## theta   0.52    0.00 0.07   0.38   0.47   0.52   0.57   0.66  5195    1
## lp__  -35.14    0.01 0.74 -37.22 -35.30 -34.85 -34.67 -34.62  5631    1
## 
## Samples were drawn using NUTS(diag_e) at Mon Nov 19 21:35:55 2018.
## 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).

Fuzzy caterpillars are good.

stan_trace(fit)

Stan uses ggplot2.

stan_hist(fit, bins=20) + xlim(0,1)

What’s the posterior probability that \(\theta < 0.5\)?

samples <- extract(fit)
mean(samples$theta < 0.5)

## [1] 0.3860667

# compare to analytic solution
pbeta(0.5, shape1=10+6, shape2=10+4)

## [1] 0.3555356

Your turn

set.seed(23)
ncoins <- 100
true_theta <- rbeta(ncoins, 20, 50)
N <- rep(50, ncoins)
Z <- rbinom(ncoins, size=N, prob=true_theta)

data {
    int n;      // number of coins
    int N[n];   // number of flips
    int Z[n];   // number of heads
}
parameters {
    // probability of heads
    real<lower=0,upper=1> theta[n];
    real<lower=0,upper=100> alpha;
    real<lower=0,upper=100> beta;
}
model {
    Z ~ binomial(N, theta);
    theta ~ beta(alpha, beta);
    // uniform priors 'go without saying'
    // alpha ~ uniform(0, 100);
    // beta ~ uniform(0, 100);
}

Baseball

We have a dataset of batting averages of baseball players, having

1. name
2. position
3. number of at bats
4. number of hits

batting <- read.csv("BattingAverage.csv", header=TRUE)
head(batting)

##          Player     PriPos Hits AtBats PlayerNumber PriPosNumber
## 1 Fernando Abad    Pitcher    1      7            1            1
## 2   Bobby Abreu Left Field   53    219            2            7
## 3    Tony Abreu   2nd Base   18     70            3            4
## 4 Dustin Ackley   2nd Base  137    607            4            4
## 5    Matt Adams   1st Base   21     86            5            3
## 6 Nathan Adcock    Pitcher    0      1            6            1

The overall batting average of the 948 players is 0.2546597.

Here is the average by position.

batting %>% group_by(PriPos) %>% 
    summarise(num=n(), BatAvg=sum(Hits)/sum(AtBats)) %>% 
    select(PriPos, num, BatAvg)

## # A tibble: 9 x 3
##   PriPos         num BatAvg
##   <fct>        <int>  <dbl>
## 1 1st Base        81  0.259
## 2 2nd Base        72  0.256
## 3 3rd Base        75  0.265
## 4 Catcher        103  0.247
## 5 Center Field    67  0.264
## 6 Left Field     103  0.259
## 7 Pitcher        324  0.129
## 8 Right Field     60  0.264
## 9 Shortstop       63  0.255

Questions?

1. What’s the overall batting average?

2. Do some positions tend to be better batters?

3. How much variation is there?

Everyone is the same

first_model <- "
data {
    int N;
    int hits[N];
    int at_bats[N];
}
parameters {
    real<lower=0, upper=1> theta;
}
model {
    hits ~ binomial(at_bats, theta);
    theta ~ beta(1, 1);
} "
first_fit <- stan(model_code=first_model, chains=3, iter=1000,
                  data=list(N=nrow(batting),
                            hits=batting$Hits,
                            at_bats=batting$AtBats))

Every pitcher is the same

pos_model <- "
data {
    int N;
    int hits[N];
    int at_bats[N];
    int npos; // number of positions
    int position[N];
}
parameters {
    real<lower=0, upper=1> theta[npos];
}
model {
    real theta_vec[N];
    for (k in 1:N) {
        theta_vec[k] = theta[position[k]];
    }
    hits ~ binomial(at_bats, theta_vec);
    theta ~ beta(1, 1);
} "
pos_fit <- stan(model_code=pos_model, chains=3, iter=1000,
                  data=list(N=nrow(batting),
                            hits=batting$Hits,
                            at_bats=batting$AtBats,
                            npos=nlevels(batting$PriPos),
                            position=as.numeric(batting$PriPos)))

Your turn : Every individual different.

int x;       // an integer
int y[10];   // ten integers
real z;      // a number
real z[2,5]; // a 2x5 array of numbers

vector u[10];      // length 10 vector
matrix v[10,10];   // 10x10 matrix
vector[10] w[10];  // ten length 10 vectors

• Stan uses MCMC to sample from the posterior

• which lets you fit realistic models.