Bayesian simulation pipelines with jagstargets

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 many simulated datasets. Examples:

1. Validate the implementation of a Bayesian model, using simulation to determine how reliably the model estimates the parameters under known data-generating scenarios.
2. Simulate a randomized controlled experiment to explore frequentist properties such as power and Type I error.

This vignette focuses on (1). The goal of this particular example 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. 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.

Consider a simple regression model with a continuous response y with a covariate x.

$$ \begin{aligned} y_i &\stackrel{\text{iid}}{\sim} \text{Normal}(\beta_1 + x_i \beta_2, 1) \\ \beta_1, \beta_2 &\stackrel{\text{iid}}{\sim} \text{Normal}(0, 1) \end{aligned} $$

We write this model in a JAGS model file.

lines <- "model {
  for (i in 1:n) {
    y[i] ~ dnorm(beta[1] + x[i] * beta[2], 1)
  }
  for (i in 1:2) {
    beta[i] ~ dnorm(0, 1)
  }
}"
writeLines(lines, "model.jags")

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. jagstargets 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 20 datasets (5 batches with 4 replications each) and run the model on each of the 20 datasets.¹ By default, each of the 20 model runs computes 3 MCMC chains with 2000 MCMC iterations each (including burn-in) and you can adjust with the n.chains and n.iter arguments of tar_jags_rep_summary().

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

generate_data <- function(n = 10L) {
  beta <- stats::rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- stats::rnorm(n, beta[1] + x * beta[2], 1)
  # Elements of .join_data get joined on to the .join_data column
  # in the summary output next to the model parameters
  # with the same names.
  .join_data <- list(beta = beta)
  list(n = n, x = x, y = y, .join_data = .join_data)
}

list(
  tar_jags_rep_summary(
    model,
    "model.jags",
    data = generate_data(),
    parameters.to.save = "beta",
    batches = 5, # Number of branch targets.
    reps = 4, # Number of model reps per branch target.
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    )
  )
)

We now have a pipeline that runs the model 10 times: 5 batches (branch targets) with 4 replications per batch.

tar_visnetwork()

Run the computation with tar_make()

tar_make()

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 the .join_data list we included in generate_data(), our output has a .join_data column with the true values of the parameters in our simulations.

tar_load(model)
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) %>%
  dplyr::summarize(coverage = mean(q2.5 < .join_data & .join_data < q97.5))

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

# _targets.R
# packages needed to define the pipeline:
library(targets)
library(jagstargets)

tar_option_set(
  packages = "dplyr", # packages needed to run the pipeline
  memory = "transient", # memory efficiency
  garbage_collection = TRUE # memory efficiency
)

generate_data <- function(n = 10L) {
  beta <- stats::rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- stats::rnorm(n, beta[1] + x * beta[2], 1)
  # Elements of .join_data get joined on to the .join_data column
  # in the summary output next to the model parameters
  # with the same names.
  .join_data <- list(beta = beta)
  list(n = n, x = x, y = y, .join_data = .join_data)
}

list(
  tar_jags_rep_summary(
    model,
    "model.jags",
    data = generate_data(),
    parameters.to.save = "beta",
    batches = 5, # Number of branch targets.
    reps = 4, # Number of model reps per branch target.
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    )
  ),
  tar_target(
    coverage,
    model %>%
      group_by(variable) %>%
      summarize(
        coverage = mean(q2.5 < .join_data & .join_data < q97.5),
        .groups = "drop"
      )
  )
)

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()

tar_read(coverage)

lines <- "model {
  for (i in 1:n) {
    y[i] ~ dnorm(beta[1] + x[i] * x[i] * beta[2], 1) # Regress on x^2, not x.
  }
  for (i in 1:2) {
    beta[i] ~ dnorm(0, 1)
  }
}"
writeLines(lines, "model2.jags")

# _targets.R
# packages needed to define the pipeline:
library(targets)
library(jagstargets)

tar_option_set(
  packages = "dplyr", # packages needed to run the pipeline
  memory = "transient", # memory efficiency
  garbage_collection = TRUE # memory efficiency
)

generate_data <- function(n = 10L) {
  beta <- stats::rnorm(n = 2, mean = 0, sd = 1)
  x <- seq(from = -1, to = 1, length.out = n)
  y <- stats::rnorm(n, beta[1] + x * beta[2], 1)
  # Elements of .join_data get joined on to the .join_data column
  # in the summary output next to the model parameters
  # with the same names.
  .join_data <- list(beta = beta)
  list(n = n, x = x, y = y, .join_data = .join_data)
}

list(
  tar_jags_rep_summary(
    model,
    c("model.jags", "model2.jags"), # another model
    data = generate_data(),
    parameters.to.save = "beta",
    batches = 5,
    reps = 4,
    variables = "beta",
    summaries = list(
      ~posterior::quantile2(.x, probs = c(0.025, 0.975))
    )
  ),
  tar_target(
    coverage,
    model %>%
      group_by(.name) %>%
      summarize(coverage = mean(q2.5 < .join_data & .join_data < q97.5))
  )
)

tar_visnetwork()