Bayesian simulation pipelines with stantargets

Background

The introductory vignette vignette caters to Bayesian data analysis workflows with few datasets to analyze. However, it is sometimes desirable to run one or more Bayesian models repeatedly across multiple simulated datasets. Examples:

  1. Validate the implementation of a Bayesian model using simulation.
  2. Simulate a randomized controlled experiment to explore frequentist properties such as power and Type I error.

This vignette focuses on (1).

Example project

Visit https://github.com/wlandau/stantargets-example-validation for an example project based on this vignette. The example has an RStudio Cloud workspace which allows you to run the project in a web browser.

Interval-based model validation pipeline

This particular example uses the concept of calibration that Bob Carpenter explains here (Carpenter 2017). The goal is to simulate multiple datasets from the model below, analyze each dataset, and assess how often the estimated posterior intervals cover the true parameters from the prior predictive simulations. If coverage is no systematically different from nominal, this is evidence that the model was implemented correctly. The quantile method by Cook, Gelman, and Rubin (2006) generalizes this concept, and simulation-based calibration (Talts et al. 2020) generalizes further. The interval-based technique featured in this vignette is not as robust as SBC, but it may be more expedient for large models because it does not require visual inspection of multiple histograms. See a later section in this vignette for an example of simulation-based calibration on this same model.

lines <- "data {
  int <lower = 1> n;
  vector[n] x;
  vector[n] y;
}
parameters {
  vector[2] beta;
}
model {
  y ~ normal(beta[1] + x * beta[2], 1);
  beta ~ normal(0, 1);
}"
writeLines(lines, "model.stan")

Next, we define a pipeline to simulate multiple datasets and fit each dataset with the model. In our data-generating function, we put the true parameter values of each simulation in a special .join_data list. stantargets will automatically join the elements of .join_data to the correspondingly named variables in the summary output. This will make it super easy to check how often our posterior intervals capture the truth. As for scale, generate 10 datasets (5 batches with 2 replications each) and run the model on each of the 10 datasets.1 By default, each of the 10 model runs computes 4 MCMC chains with 2000 MCMC iterations each (including burn-in) and you can adjust with the chains, iter_sampling, and iter_warmup arguments of tar_stan_mcmc_rep_summary().

# _targets.R
library(targets)
library(stantargets)
options(crayon.enabled = FALSE)
# Use computer memory more sparingly:
tar_option_set(memory = "transient", garbage_collection = TRUE)

simulate_data <- function(n = 10L) {
  beta <- rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- rnorm(n, beta[1] + x * beta[2], 1)
  list(
    n = n,
    x = x,
    y = y,
    .join_data = list(beta = beta)
  )
}

list(
  tar_stan_mcmc_rep_summary(
    model,
    "model.stan",
    simulate_data(), # Runs once per rep.
    batches = 5, # Number of branch targets.
    reps = 2, # Number of model reps per branch target.
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    ),
    stdout = R.utils::nullfile(),
    stderr = R.utils::nullfile()
  )
)

We now have a pipeline that runs the model 10 times: 5 batches (branch targets) with 2 replications per batch.

tar_visnetwork()

Run the computation with tar_make()

tar_make()
#> ▶ dispatched target model_batch
#> ● completed target model_batch [0 seconds, 99 bytes]
#> ▶ dispatched target model_file_model
#> ● completed target model_file_model [37.309 seconds, 2.637 megabytes]
#> ▶ dispatched branch model_data_5fcdec5f855f2d9c
#> ● completed branch model_data_5fcdec5f855f2d9c [0.005 seconds, 509 bytes]
#> ▶ dispatched branch model_data_b6c9a18333c6a8ca
#> ● completed branch model_data_b6c9a18333c6a8ca [0.001 seconds, 507 bytes]
#> ▶ dispatched branch model_data_5db4354944466148
#> ● completed branch model_data_5db4354944466148 [0.001 seconds, 507 bytes]
#> ▶ dispatched branch model_data_4a40cb783277d5dc
#> ● completed branch model_data_4a40cb783277d5dc [0 seconds, 508 bytes]
#> ▶ dispatched branch model_data_104af6d505e730d6
#> ● completed branch model_data_104af6d505e730d6 [0.001 seconds, 508 bytes]
#> ● completed pattern model_data 
#> ▶ dispatched branch model_model_50b3d9bcb9189fef
#> ● completed branch model_model_50b3d9bcb9189fef [1.516 seconds, 1.269 kilobytes]
#> ▶ dispatched branch model_model_93bc2c2a4b8dc29f
#> ● completed branch model_model_93bc2c2a4b8dc29f [1.305 seconds, 1.269 kilobytes]
#> ▶ dispatched branch model_model_e2ab729f4fa1dd45
#> ● completed branch model_model_e2ab729f4fa1dd45 [1.307 seconds, 1.269 kilobytes]
#> ▶ dispatched branch model_model_5871bb9227fbbf93
#> ● completed branch model_model_5871bb9227fbbf93 [1.306 seconds, 1.269 kilobytes]
#> ▶ dispatched branch model_model_820c742ab2ba1134
#> ● completed branch model_model_820c742ab2ba1134 [1.302 seconds, 1.269 kilobytes]
#> ● completed pattern model_model 
#> ▶ dispatched target model
#> ● completed target model [0 seconds, 2.897 kilobytes]
#> ▶ ended pipeline [45.636 seconds]

The result is an aggregated data frame of summary statistics, where the .rep column distinguishes among individual replicates. We have the posterior intervals for beta in columns q2.5 and q97.5. And thanks to .join_data in simulate_data(), there is a special .join_data column in the output to indicate the true value of each parameter from the simulation.

tar_load(model)
model
#> # A tibble: 20 × 9
#>    variable    q2.5   q97.5 .join_data .rep       .dataset_id  .seed .file .name
#>    <chr>      <dbl>   <dbl>      <dbl> <chr>      <chr>        <int> <chr> <chr>
#>  1 beta[1]   0.385   1.59       0.449  be2f1457d… model_data… 5.71e8 mode… model
#>  2 beta[2]  -1.74    0.0192    -0.712  be2f1457d… model_data… 5.71e8 mode… model
#>  3 beta[1]   1.54    2.68       2.18   ef03cb821… model_data… 1.03e9 mode… model
#>  4 beta[2]  -0.836   0.886      0.315  ef03cb821… model_data… 1.03e9 mode… model
#>  5 beta[1]   0.552   1.75       0.892  1a5d27aea… model_data… 1.92e9 mode… model
#>  6 beta[2]   1.41    3.17       1.98   1a5d27aea… model_data… 1.92e9 mode… model
#>  7 beta[1]   1.33    2.54       1.33   aec745e92… model_data… 1.95e9 mode… model
#>  8 beta[2]  -0.911   0.808     -0.454  aec745e92… model_data… 1.95e9 mode… model
#>  9 beta[1]  -0.0580  1.09       0.0642 5972613fb… model_data… 7.78e8 mode… model
#> 10 beta[2]   0.0102  1.74       1.12   5972613fb… model_data… 7.78e8 mode… model
#> 11 beta[1]  -0.105   1.06       0.403  9274f1b95… model_data… 1.90e9 mode… model
#> 12 beta[2]  -0.993   0.747      0.868  9274f1b95… model_data… 1.90e9 mode… model
#> 13 beta[1]  -1.07    0.124     -0.262  2624ff163… model_data… 3.01e8 mode… model
#> 14 beta[2]   0.804   2.54       1.66   2624ff163… model_data… 3.01e8 mode… model
#> 15 beta[1]  -0.381   0.790     -0.548  94540fa58… model_data… 5.21e8 mode… model
#> 16 beta[2]   0.697   2.43       1.39   94540fa58… model_data… 5.21e8 mode… model
#> 17 beta[1]  -0.994   0.132     -0.120  fd6c8fcb8… model_data… 1.10e9 mode… model
#> 18 beta[2]  -0.156   1.66       0.931  fd6c8fcb8… model_data… 1.10e9 mode… model
#> 19 beta[1]  -2.06   -0.879     -1.75   562b22793… model_data… 1.34e9 mode… model
#> 20 beta[2]  -1.92   -0.186     -0.474  562b22793… model_data… 1.34e9 mode… model

Now, let’s assess how often the estimated 95% posterior intervals capture the true values of beta. If the model is implemented correctly, the coverage value below should be close to 95%. (Ordinarily, we would increase the number of batches and reps per batch and run batches in parallel computing.)

library(dplyr)
model %>%
  group_by(variable) %>%
  summarize(coverage = mean(q2.5 < .join_data & .join_data < q97.5))
#> # A tibble: 2 × 2
#>   variable coverage
#>   <chr>       <dbl>
#> 1 beta[1]       0.8
#> 2 beta[2]       0.9

For maximum reproducibility, we should express the coverage assessment as a custom function and a target in the pipeline.

# _targets.R
library(targets)
library(stantargets)

simulate_data <- function(n = 10L) {
  beta <- rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- rnorm(n, beta[1] + x * beta[2], 1)
  list(
    n = n,
    x = x,
    y = y,
    .join_data = list(beta = beta)
  )
}

list(
  tar_stan_mcmc_rep_summary(
    model,
    "model.stan",
    simulate_data(),
    batches = 5, # Number of branch targets.
    reps = 2, # Number of model reps per branch target.
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    ),
    stdout = R.utils::nullfile(),
    stderr = R.utils::nullfile()
  ),
  tar_target(
    coverage,
    model %>%
      group_by(variable) %>%
      summarize(coverage = mean(q2.5 < .join_data & .join_data < q97.5))
  )
)

The new coverage target should the only outdated target, and it should be connected to the upstream model target.

tar_visnetwork()

When we run the pipeline, only the coverage assessment should run. That way, we skip all the expensive computation of simulating datasets and running MCMC multiple times.

tar_make()
#> ✔ skipping targets (1 so far)...
#> ▶ dispatched target coverage
#> ● completed target coverage [0.009 seconds, 179 bytes]
#> ▶ ended pipeline [0.466 seconds]
tar_read(coverage)
#> # A tibble: 2 × 2
#>   variable coverage
#>   <chr>       <dbl>
#> 1 beta[1]       0.8
#> 2 beta[2]       0.9

Multiple models

tar_stan_rep_mcmc_summary() and similar functions allow you to supply multiple Stan models. If you do, each model will share the the same collection of datasets, and the .dataset_id column of the model target output allows for custom analyses that compare different models against each other. Suppose we have a new model, model2.stan.

lines <- "data {
  int <lower = 1> n;
  vector[n] x;
  vector[n] y;
}
parameters {
  vector[2] beta;
}
model {
  y ~ normal(beta[1] + x * x * beta[2], 1); // Regress on x^2 instead of x.
  beta ~ normal(0, 1);
}"
writeLines(lines, "model2.stan")

To set up the simulation workflow to run on both models, we add model2.stan to the stan_files argument of tar_stan_rep_mcmc_summary(). And in the coverage summary below, we group by .name to compute a coverage statistic for each model.

# _targets.R
library(targets)
library(stantargets)

simulate_data <- function(n = 10L) {
  beta <- rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- rnorm(n, beta[1] + x * beta[2], 1)
  list(
    n = n,
    x = x,
    y = y,
    .join_data = list(beta = beta)
  )
}

list(
  tar_stan_mcmc_rep_summary(
    model,
    c("model.stan", "model2.stan"), # another model
    simulate_data(),
    batches = 5,
    reps = 2,
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    ),
    stdout = R.utils::nullfile(),
    stderr = R.utils::nullfile()
  ),
  tar_target(
    coverage,
    model %>%
      group_by(.name, variable) %>%
      summarize(coverage = mean(q2.5 < .join_data & .join_data < q97.5))
  )
)

In the graph below, notice how targets model_model and model_model2 are both connected to model_data upstream. Downstream, model is equivalent to dplyr::bind_rows(model_model, model_model2), and it will have special columns .name and .file to distinguish among all the models.

tar_visnetwork()

Simulation-based calibration

This section explores a more rigorous validation study which adopts the proper simulation-based calibration (SBC) method from (Talts et al. 2020). To use this method, we need a function that generates rank statistics from a simulated dataset and a data frame of posterior draws. If the model is implemented correctly, these rank statistics will be uniformly distributed for each model parameter. Our function will use the calculate_ranks_draws_matrix() function from the SBC R package (Kim et al. 2022).

get_ranks <- function(data, draws) {
  draws <- select(draws, starts_with(names(data$.join_data)))
  truth <- map_dbl(
    names(draws),
    ~eval(parse(text = .x), envir = data$.join_data)
  )
  out <- SBC::calculate_ranks_draws_matrix(truth, as_draws_matrix(draws))
  as_tibble(as.list(out))
}

To demonstrate this function, we simulate a dataset,

simulate_data <- function(n = 10L) {
  beta <- rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- rnorm(n, beta[1] + x * beta[2], 1)
  list(
    n = n,
    x = x,
    y = y,
    .join_data = list(beta = beta)
  )
}

data <- simulate_data()

str(data)
#> List of 4
#>  $ n         : int 10
#>  $ x         : num [1:10] -1 -0.778 -0.556 -0.333 -0.111 ...
#>  $ y         : num [1:10] -0.803 -1.157 1.011 -0.302 0.825 ...
#>  $ .join_data:List of 1
#>   ..$ beta: num [1:2] 0.0884 0.2284

we make up a hypothetical set of posterior draws,

draws <- tibble(`beta[1]` = rnorm(100), `beta[2]` = rnorm(100))

draws
#> # A tibble: 100 × 2
#>    `beta[1]` `beta[2]`
#>        <dbl>     <dbl>
#>  1     0.770    1.23  
#>  2     0.540    0.245 
#>  3     1.18     0.0918
#>  4     0.578   -0.345 
#>  5    -2.01    -0.849 
#>  6     1.65     1.26  
#>  7     1.01     1.02  
#>  8    -1.85     0.174 
#>  9    -0.210   -0.656 
#> 10     0.672   -0.584 
#> # ℹ 90 more rows

and we call get_ranks() to get the SBC rank statistics for each model parameter.

library(dplyr)
library(posterior)
library(purrr)
get_ranks(data = data, draws = draws)
#> # A tibble: 1 × 2
#>   `beta[1]` `beta[2]`
#>       <dbl>     <dbl>
#> 1        47        59

To put this into practice in a pipeline, we supply the symbol get_ranks to the transform argument of tar_stan_mcmc_rep_draws(). That way, instead of a full set of draws, each replication will return only the output of get_ranks() on those draws (plus a few helper columns). If supplied, the transform argument of tar_stan_mcmc_rep_draws() must be the name of a function in the pipeline. This function must accept arguments data and draws, and it must return a data frame.

# _targets.R
library(targets)
library(stantargets)

tar_option_set(packages = c("dplyr", "posterior", "purrr", "tibble"))

simulate_data <- function(n = 10L) {
  beta <- rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- rnorm(n, beta[1] + x * beta[2], 1)
  list(
    n = n,
    x = x,
    y = y,
    .join_data = list(beta = beta)
  )
}

get_ranks <- function(data, draws) {
  draws <- select(draws, starts_with(names(data$.join_data)))
  truth <- map_dbl(
    names(draws),
    ~eval(parse(text = .x), envir = data$.join_data)
  )
  out <- SBC::calculate_ranks_draws_matrix(truth, as_draws_matrix(draws))
  as_tibble(as.list(out))
}

list(
  tar_stan_mcmc_rep_draws(
    model,
    c("model.stan"),
    simulate_data(),
    batches = 5,
    reps = 2,
    variables = "beta",
    stdout = R.utils::nullfile(),
    stderr = R.utils::nullfile(),
    transform = get_ranks # Supply the transform to get SBC ranks.
  )
)

Our new function get_ranks() is a dependency of one of our downstream targets, so any changes to get_ranks() will force the results to refresh in the next run of the pipeline.

tar_visnetwork()

Let’s run the pipeline to compute a set of rank statistics for each simulated dataset.

tar_make()
#> ✔ skipping targets (1 so far)...
#> ▶ dispatched branch model_model_50b3d9bcb9189fef
#> ● completed branch model_model_50b3d9bcb9189fef [1.629 seconds, 875 bytes]
#> ▶ dispatched branch model_model_93bc2c2a4b8dc29f
#> ● completed branch model_model_93bc2c2a4b8dc29f [1.315 seconds, 876 bytes]
#> ▶ dispatched branch model_model_e2ab729f4fa1dd45
#> ● completed branch model_model_e2ab729f4fa1dd45 [1.312 seconds, 876 bytes]
#> ▶ dispatched branch model_model_5871bb9227fbbf93
#> ● completed branch model_model_5871bb9227fbbf93 [1.31 seconds, 876 bytes]
#> ▶ dispatched branch model_model_820c742ab2ba1134
#> ● completed branch model_model_820c742ab2ba1134 [1.31 seconds, 876 bytes]
#> ● completed pattern model_model 
#> ▶ ended pipeline [7.89 seconds]

We have a data frame of rank statistics with one row per simulation rep and one column per model parameter.

tar_load(model_model)

model_model
#> # A tibble: 10 × 7
#>    `beta[1]` `beta[2]` .rep             .dataset_id            .seed .file .name
#>        <dbl>     <dbl> <chr>            <chr>                  <int> <chr> <chr>
#>  1       152      2550 e4c40d64f73a2645 model_data_5fcdec5f8… 5.71e8 mode… model
#>  2      2410      2890 e8cb87f562257cd0 model_data_5fcdec5f8… 1.03e9 mode… model
#>  3       789       976 1cce3b1262ed409b model_data_b6c9a1833… 1.92e9 mode… model
#>  4       100       764 872064d78a80c709 model_data_b6c9a1833… 1.95e9 mode… model
#>  5       263      2834 dc5816106f55e6f2 model_data_5db435494… 7.78e8 mode… model
#>  6      1629      3946 2b1a2bb9e998e263 model_data_5db435494… 1.90e9 mode… model
#>  7      3054      1933 12b5175aee17f142 model_data_4a40cb783… 3.01e8 mode… model
#>  8        19      1443 549bd80f183be212 model_data_4a40cb783… 5.21e8 mode… model
#>  9      3405      2547 0b892d9d8c225f3e model_data_104af6d50… 1.10e9 mode… model
#> 10       715      3610 29973d97ee2f3be6 model_data_104af6d50… 1.34e9 mode… model

If the model is implemented correctly, then each the rank statistics each model parameter should be uniformly distributed. In practice, you may need thousands of simulation reps to make a judgment.

library(ggplot2)
library(tidyr)
model_model %>%
  pivot_longer(
    starts_with("beta"),
    names_to = "parameter",
    values_to = "ranks"
  ) %>%
  ggplot(.) +
  geom_histogram(aes(x = ranks), bins = 10) +
  facet_wrap(~parameter) +
  theme_gray(12)

References

Carpenter, Bob. 2017. “Bayesian Posteriors are Calibrated by Definition.” https://statmodeling.stat.columbia.edu/2017/04/12/bayesian-posteriors-calibrated/.
Cook, Samantha R., Andrew Gelman, and Donald B. Rubin. 2006. “Validation of Software for Bayesian Models Using Posterior Quantiles.” Journal of Computational and Graphical Statistics 15 (3): 675–92. http://www.jstor.org/stable/27594203.
Kim, Shinyoung, Hyunji Moon, Martin Modrák, and Teemu Säilynoja. 2022. SBC: Simulation Based Calibration for Rstan/Cmdstanr Models.
Talts, Sean, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. 2020. “Validating Bayesian Inference Algorithms with Simulation-Based Calibration.” https://arxiv.org/abs/1804.06788.

  1. Internally, each batch is a dynamic branch target, and the number of replications determines the amount of work done within a branch. In the general case, batching is a way to find the right compromise between target-specific overhead and the horizontal scale of the pipeline.↩︎