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:
- Validate the implementation of a Bayesian model, using simulation to determine how reliably the model estimates the parameters under known data-generating scenarios.
- 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.
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)
modelNow, 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.
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)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.