# Simulation Based Calibration

#### Stan Development Team

#### 2023-09-07

Here is a Stan program for a beta-binomial model

    data {
      int<lower = 1> N;
      real<lower = 0> a;
      real<lower = 0> b;
    }
    transformed data { // these adhere to the conventions above
      real pi_ = beta_rng(a, b);
      int y = binomial_rng(N, pi_);
    }
    parameters {
      real<lower = 0, upper = 1> pi;
    }
    model {
      target += beta_lpdf(pi | a, b);
      target += binomial_lpmf(y | N, pi);
    }
    generated quantities { // these adhere to the conventions above
      int y_ = y;
      vector[1] pars_;
      int ranks_[1] = {pi > pi_};
      vector[N] log_lik;
      pars_[1] = pi_;
      for (n in 1:y) log_lik[n] = bernoulli_lpmf(1 | pi);
      for (n in (y + 1):N) log_lik[n] = bernoulli_lpmf(0 | pi);
    }

Notice that it adheres to the following conventions:

- Realizations of the unknown parameters are drawn in the `transformed data` block are postfixed with an underscore, such as `pi_`. These are considered the “true” parameters being estimated by the corresponding symbol declared in the `parameters` block, which have the same names except for the trailing underscore, such as `pi`.
- The realizations of the unknown parameters are then conditioned on when drawing from the prior predictive distribution in `transformed data` block, which in this case is `int y = binomial_rng(N, pi_);`. To avoid confusion, `y` does not have a training underscore.
- The realizations of the unknown parameters are copied into a `vector` in the `generated quantities` block named `pars_`
- The realizations from the prior predictive distribution are copied into an object (of the same type) in the `generated quantities` block named `y_. This is optional.
- The `generated quantities` block contains an integer array named `ranks_` whose only values are zero or one, depending on whether the realization of a parameter from the posterior distribution exceeds the corresponding “true” realization, which in this case is `ranks_[1] = {pi > pi_};`. These are not actually “ranks” but can be used afterwards to reconstruct (thinned) ranks.
- The `generated quantities` block contains a vector named `log_lik` whose values are the contribution to the log-likelihood by each observation. In this case, the “observations” are the implicit successes and failures that are aggregated into a binomial likelihood. This is optional but facilitates calculating the Pareto k shape parameter estimates that indicate whether the posterior distribution is sensitive to particular observations.

Assuming the above is compile to a code `stanmodel` named `beta_binomial`, we can then call the `sbc` function

    output <- sbc(beta_binomial, data = list(N = 10, a = 1, b = 1), M = 500, refresh = 0)

At which point, we can then call

    print(output)

    ## 0 chains had divergent transitions after warmup

    plot(output, bins = 10) # it is best to specify the bins argument yourself

# References

Talts, S., Betancourt, M., Simpson, D., Vehtari, A., and Gelman, A. (2018). Validating Bayesian Inference Algorithms with Simulation-Based Calibration. arXiv preprint arXiv:1804.06788