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31.3 Poststratification in Stan

The maximum likelihood and Bayesian estimates can be handled with the same Stan program. The model of individual votes is collapsed to a binomial, where \(A_j\) is the number of voters from group \(j\), \(a_j\) is the number of positive responses from group \(j\), and \(N_j\) is the size of group \(j\) in the population.

data {
  int<lower=1> J;
  array[J] int<lower=0> A; 
  array[J] int<lower=0> a;
  vector<lower=0>[J] N;
}
parameters {
  vector<lower=0, upper=1>[J] theta;
}
model {
  a ~ binomial(A, theta);
}
generated quantities {t
  real<lower=0, upper=1> phi = dot(N, theta) / sum(N);
}

The likelihood is vectorized, and implicitly sums over the \(j\). The prior is implicitly uniform on \((0, 1),\) the support of \(\theta.\) The summation is computed using a dot product and the sum function, which is why N was declared as a vector rather than as an array of integers.