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:

Validate the implementation of a Bayesian model using simulation.
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()
#> • start target model_batch
#> • built target model_batch [0.001 seconds]
#> • start target model_file_model
#> • built target model_file_model [8.296 seconds]
#> • start branch model_data_b0b9380a
#> • built branch model_data_b0b9380a [0.008 seconds]
#> • start branch model_data_ffcdb73c
#> • built branch model_data_ffcdb73c [0.003 seconds]
#> • start branch model_data_b968a03a
#> • built branch model_data_b968a03a [0.003 seconds]
#> • start branch model_data_f8763cb2
#> • built branch model_data_f8763cb2 [0.003 seconds]
#> • start branch model_data_0bfdabdc
#> • built branch model_data_0bfdabdc [0.003 seconds]
#> • built pattern model_data
#> • start branch model_model_5d061b58
#> • built branch model_model_5d061b58 [1.618 seconds]
#> • start branch model_model_a9336683
#> • built branch model_model_a9336683 [1.36 seconds]
#> • start branch model_model_bde6a6d6
#> • built branch model_model_bde6a6d6 [1.364 seconds]
#> • start branch model_model_384f982f
#> • built branch model_model_384f982f [1.359 seconds]
#> • start branch model_model_0d59666a
#> • built branch model_model_0d59666a [1.366 seconds]
#> • built pattern model_model
#> • start target model
#> • built target model [0 seconds]
#> • end pipeline [16.986 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]   0.0318  1.24       0.751  99bfcc99 model_data_b… 1.49e9 mode… model
#>  2 beta[2]  -1.04    0.734     -0.592  99bfcc99 model_data_b… 1.49e9 mode… model
#>  3 beta[1]   0.472   1.63       1.19   051ce394 model_data_b… 2.04e9 mode… model
#>  4 beta[2]  -0.844   0.911      0.0635 051ce394 model_data_b… 2.04e9 mode… model
#>  5 beta[1]   1.35    2.52       2.15   4258cff2 model_data_f… 4.83e8 mode… model
#>  6 beta[2]  -1.94   -0.152     -1.71   4258cff2 model_data_f… 4.83e8 mode… model
#>  7 beta[1]  -0.230   0.946      0.271  96e3f8e9 model_data_f… 6.64e8 mode… model
#>  8 beta[2]  -1.73    0.0476    -0.407  96e3f8e9 model_data_f… 6.64e8 mode… model
#>  9 beta[1]   0.348   1.51       0.758  9e8ac8fd model_data_b… 1.33e9 mode… model
#> 10 beta[2]  -0.794   0.962      0.426  9e8ac8fd model_data_b… 1.33e9 mode… model
#> 11 beta[1]  -0.864   0.322     -0.541  289ec442 model_data_b… 8.94e8 mode… model
#> 12 beta[2]  -1.24    0.500     -0.251  289ec442 model_data_b… 8.94e8 mode… model
#> 13 beta[1]  -0.194   0.998      0.618  7b68a6d3 model_data_f… 3.71e7 mode… model
#> 14 beta[2]  -0.551   1.17      -0.245  7b68a6d3 model_data_f… 3.71e7 mode… model
#> 15 beta[1]  -0.913   0.254     -0.160  fb1ab4a4 model_data_f… 1.33e9 mode… model
#> 16 beta[2]  -2.52   -0.751     -1.07   fb1ab4a4 model_data_f… 1.33e9 mode… model
#> 17 beta[1]  -2.79   -1.63      -1.81   7640ef0e model_data_0… 5.73e8 mode… model
#> 18 beta[2]  -2.44   -0.673     -1.22   7640ef0e model_data_0… 5.73e8 mode… model
#> 19 beta[1]   0.483   1.64       1.43   c9c1f653 model_data_0… 1.83e9 mode… model
#> 20 beta[2]  -1.98   -0.294     -1.19   c9c1f653 model_data_0… 1.83e9 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]         1
#> 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
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()
#> ✔ 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 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 [0.014 seconds]
#> • end pipeline [0.489 seconds]
#>

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

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,

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] 0.654 1.487 0.339 -0.669 0.74 ...
#>  $ .join_data:List of 1
#>   ..$ beta: num [1:2] 1.12 1.87

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    0.0384   -1.38  
#>  2    0.246     0.177 
#>  3    0.242    -0.121 
#>  4   -0.226    -0.780 
#>  5    0.367     0.473 
#>  6    0.314     0.408 
#>  7    0.184     0.318 
#>  8    0.981     0.0144
#>  9    0.220    -1.25  
#> 10   -0.462    -1.34  
#> # … with 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        87        98

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()
#> ✔ 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
#> • start branch model_model_5d061b58
#> • built branch model_model_5d061b58 [1.663 seconds]
#> • start branch model_model_a9336683
#> • built branch model_model_a9336683 [1.371 seconds]
#> • start branch model_model_bde6a6d6
#> • built branch model_model_bde6a6d6 [1.367 seconds]
#> • start branch model_model_384f982f
#> • built branch model_model_384f982f [1.371 seconds]
#> • start branch model_model_0d59666a
#> • built branch model_model_0d59666a [1.361 seconds]
#> • built pattern model_model
#> • end pipeline [8.169 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      2600       658 7c991244 model_data_b0b9380a_1 1490816252 model.st… model
#>  2      2616      2049 e4141b79 model_data_b0b9380a_2 2036930449 model.st… model
#>  3      3098       298 b4f14839 model_data_ffcdb73c_1  483483223 model.st… model
#>  4      1646      3284 2c7051cc model_data_ffcdb73c_2  664499179 model.st… model
#>  5      1122      3083 4d2b0770 model_data_b968a03a_1 1333721666 model.st… model
#>  6       802      2402 d3199d78 model_data_b968a03a_2  894289376 model.st… model
#>  7      3031       442 5768e060 model_data_f8763cb2_1   37052332 model.st… model
#>  8      2831      3572 5589c874 model_data_f8763cb2_2 1333838785 model.st… model
#>  9      3647      3022 a8ccdc62 model_data_0bfdabdc_1  573008143 model.st… model
#> 10      3543      1781 f697a5d6 model_data_0bfdabdc_2 1825978703 model.st… 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.

Background

Example project

Interval-based model validation pipeline

Multiple models

Simulation-based calibration

References

About

Community

Resources