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()
#> • start target model_batch
#> • built target model_batch
#> • start target model_file_model
#> • built target model_file_model
#> • start branch model_data_b0b9380a
#> • built branch model_data_b0b9380a
#> • start branch model_data_ffcdb73c
#> • built branch model_data_ffcdb73c
#> • start branch model_data_b968a03a
#> • built branch model_data_b968a03a
#> • start branch model_data_f8763cb2
#> • built branch model_data_f8763cb2
#> • start branch model_data_0bfdabdc
#> • built branch model_data_0bfdabdc
#> • built pattern model_data
#> • start target model_lines_model
#> • built target model_lines_model
#> • start branch model_model_5d061b58
#> • built branch model_model_5d061b58
#> • start branch model_model_a9336683
#> • built branch model_model_a9336683
#> • start branch model_model_bde6a6d6
#> • built branch model_model_bde6a6d6
#> • start branch model_model_384f982f
#> • built branch model_model_384f982f
#> • start branch model_model_0d59666a
#> • built branch model_model_0d59666a
#> • built pattern model_model
#> • start target model
#> • built target model
#> • end pipeline: 2.748 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 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
#> # A tibble: 40 × 8
#>    variable     q2.5  q97.5 .join_data .dataset_id           .rep    .file .name
#>    <chr>       <dbl>  <dbl>      <dbl> <chr>                 <chr>   <chr> <chr>
#>  1 beta[1]   0.430    1.57       0.768 model_data_b0b9380a_1 71ac42… mode… model
#>  2 beta[2]  -0.824    0.913      0.163 model_data_b0b9380a_1 71ac42… mode… model
#>  3 beta[1]  -0.602    0.559     -0.602 model_data_b0b9380a_2 cf82d6… mode… model
#>  4 beta[2]  -0.437    1.26       1.15  model_data_b0b9380a_2 cf82d6… mode… model
#>  5 beta[1]  -0.00521  1.17       0.353 model_data_b0b9380a_3 e756b3… mode… model
#>  6 beta[2]  -0.0176   1.69       0.128 model_data_b0b9380a_3 e756b3… mode… model
#>  7 beta[1]   0.355    1.52       1.50  model_data_b0b9380a_4 bf0ac2… mode… model
#>  8 beta[2]  -1.63     0.108     -0.624 model_data_b0b9380a_4 bf0ac2… mode… model
#>  9 beta[1]   0.0702   1.27       0.631 model_data_ffcdb73c_1 bddf93… mode… model
#> 10 beta[2]  -2.91    -1.21      -2.68  model_data_ffcdb73c_1 bddf93… mode… model
#> # … with 30 more rows

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))
#> # A tibble: 2 × 2
#>   variable coverage
#>   <chr>       <dbl>
#> 1 beta[1]       0.9
#> 2 beta[2]       1

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()
#> ✔ skip target model_batch
#> ✔ skip target model_file_model
#> ✔ skip branch model_data_b0b9380a
#> ✔ skip branch model_data_ffcdb73c
#> ✔ skip branch model_data_b968a03a
#> ✔ skip branch model_data_f8763cb2
#> ✔ skip branch model_data_0bfdabdc
#> ✔ skip pattern model_data
#> ✔ skip target model_lines_model
#> ✔ skip branch model_model_5d061b58
#> ✔ skip branch model_model_a9336683
#> ✔ skip branch model_model_bde6a6d6
#> ✔ skip branch model_model_384f982f
#> ✔ skip branch model_model_0d59666a
#> ✔ skip pattern model_model
#> ✔ skip target model
#> • start target coverage
#> • built target coverage
#> • end pipeline: 0.549 seconds
tar_read(coverage)
#> # A tibble: 2 × 2
#>   variable coverage
#>   <chr>       <dbl>
#> 1 beta[1]       0.9
#> 2 beta[2]       1

Multiple models

tar_jags_rep_mcmc_summary() and similar functions allow you to supply multiple jags 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. Below, we add a new model2.jags file to the jags_files argument of tar_jags_rep_mcmc_summary(). In the coverage summary below, we group by .name to compute a coverage statistic for each model.

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))
  )
)

In the graph below, notice how targets model_model1 and model_model2 are both connected to model_data upstream. Downstream, model is equivalent to dplyr::bind_rows(model_model1, model_model2), and it will have special columns .name and .file to distinguish among all the models.

tar_visnetwork()

References

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.

Talts, Sean, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. 2020. “Validating Bayesian Inference Algorithms with Simulation-Based Calibration.”

  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.↩︎