Check whether a model is well-calibrated with respect to the prior distribution and hence possibly amenable to obtaining a posterior distribution conditional on observed data.

sbc(stanmodel, data, M, ..., save_progress, load_incomplete=FALSE)
# S3 method for sbc
plot(x, thin = 3, ...)
# S3 method for sbc
print(x, ...)

## Arguments

stanmodel An object of stanmodel-class that is first created by calling the stan_model function A named list or environment providing the data for the model, or a character vector for all the names of objects to use as data. This is the same format as in stan or sampling. The number of times to condition on draws from the prior predictive distribution Additional arguments that are passed to sampling, such as refresh = 0 when calling sbc. For the plot and print methods, the additional arguments are not used. An object produced by sbc An integer vector of length one indicating the thinning interval when plotting, which defaults to 3 If a directory is provided, stanfit objects are saved to disk making it easy to resume a partial sbc run after interruption. When save_progress is used, load whatever runs have been saved to disk and ignore argument M.

## Details

This function assumes adherence to the following conventions in the underlying Stan program:

1. Realizations of the unknown parameters are drawn in the transformed data block of the Stan program and are postfixed with an underscore, such as theta_. These are considered the “true” parameters being estimated by the corresponding symbol declared in the parameters block, which should have the same name except for the trailing underscore, such as theta.

2. The realizations of the unknown parameters are then conditioned on when drawing from the prior predictive distribution, also in the transformed data block. There is no restriction on the symbol name that holds the realizations from the prior predictive distribution but for clarity, it should not end with a trailing underscore.

3. The realizations of the unknown parameters should be copied into a vector in the generated quantities block named pars_.

4. The realizations from the prior predictive distribution should be copied into an object (of the same type) in the generated quantities block named y_. Technically, this step is optional and could be omitted to conserve RAM, but inspecting the realizations from the prior predictive distribution is a good way to judge whether the priors are reasonable.

5. The generated quantities block must contain 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, such as theta > theta_;. These are not actually "ranks" but can be used afterwards to reconstruct (thinned) ranks.

6. The generated quantities block may contain a vector named log_lik whose values are the contribution to the log-likelihood by each observation. This is optional but facilitates calculating Pareto k shape parameters to judge whether the posterior distribution is sensitive to particular observations.

Although the user can pass additional arguments to sampling through the ..., the following arguments are hard-coded and should not be passed through the ...:

1. pars = "ranks_" because nothing else needs to be stored for each posterior draw

2. include = TRUE to ensure that "ranks_" is included rather than excluded

3. chains = 1 because only one chain is run for each integer less than M

4. seed because a sequence of seeds is used across the M runs to preserve independence across runs

5. save_warmup = FALSE because the warmup realizations are not relevant

6. thin = 1 because thinning can and should be done after the Markov Chain is finished, as is done by the thin argument to the plot method in order to make the histograms consist of approximately independent realizations

Other arguments will take the default values used by sampling unless passed through the .... Specifying refresh = 0 is recommended to avoid printing a lot of intermediate progress reports to the screen. It may be necessary to pass a list to the control argument of sampling with elements adapt_delta and / or max_treedepth in order to obtain adequate results.

Ideally, users would want to see the absence of divergent transitions (which is shown by the print method) and other warnings, plus an approximately uniform histogram of the ranks for each parameter (which are shown by the plot method). See the vignette for more details.

## Value

The sbc function outputs a list of S3 class "sbc", which contains the following elements:

1. ranks A list of M matrices, each with number of rows equal to the number of saved iterations and number of columns equal to the number of unknown parameters. These matrices contain the realizations of the ranks_ object from the generated quantities block of the Stan program.

2. Y If present, a matrix of realizations from the prior predictive distribution whose rows are equal to the number of observations and whose columns are equal to M, which are taken from the y_ object in the generated quantities block of the Stan program.

3. pars A matrix of realizations from the prior distribution whose rows are equal to the number of parameters and whose columns are equal to M, which are taken from the pars_ object in the generated quantities block of the Stan program.

4. pareto_k A matrix of Pareto k shape parameter estimates or NULL if there is no log_lik symbol in the generated quantities block of the Stan program

5. sampler_params A three-dimensional array that results from combining calls to get_sampler_params for each of the M runs. The resulting matrix has rows equal to the number of post-warmup iterations, columns equal to six, and M floors. The columns are named "accept_stat__", "stepsize__", "treedepth__", "n_leapfrog__", "divergent__", and "energy__". The most important of which is "divergent__", which should be all zeros and perhaps "treedepth__", which should only rarely get up to the value of max_treedepth passed as an element of the control list to sampling or otherwise defaults to $$10$$.

The print method outputs the number of divergent transitions and returns NULL invisibly. The plot method returns a ggplot object with histograms whose appearance can be further customized.

## References

The Stan Development Team Stan Modeling Language User's Guide and Reference Manual. http://mc-stan.org.

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. https://arxiv.org/abs/1804.06788

stan_model and sampling

## Examples

scode <- "
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);
}
"