bridge_sampling: Bridge sampling for marginal likelihood

In tulpa: Templated Unified Library for Posterior Approximation in Bayesian Hierarchical Models

Bridge sampling for marginal likelihood

Description

Estimate the log marginal likelihood \log Z = \log p(y) from posterior draws via the Meng-Wong / Gronau bridge-sampling identity. Useful for Bayes factors, model comparison, and as a sanity check on Laplace-approximated marginal likelihoods.

The classical iterative scheme (Meng & Wong 1996; Gronau et al. 2017) is used with a multivariate-normal proposal fit to half of the posterior draws – the other half is used for the bridge ratio so the proposal is independent of the samples that score it. All computations are done in log-space via logsumexp for numerical stability.

Usage

bridge_sampling(
  draws,
  log_posterior,
  n_proposal = nrow(draws),
  max_iter = 1000L,
  tol = 1e-10,
  split = TRUE,
  verbose = FALSE
)

Arguments

draws	Numeric matrix of posterior draws, one parameter per column. Typically fit$draws from a tulpa Tier-1 fit.
log_posterior	Function ⁠function(theta) -> numeric⁠ returning the unnormalized log posterior density ⁠log p(theta, y)⁠ at a single parameter vector. Must accept a numeric vector of length ncol(draws) and return a finite scalar.
n_proposal	Number of proposal draws. Default nrow(draws).
max_iter	Maximum bridge iterations (default 1000).
tol	Convergence tolerance on ⁠|log r_{t+1} - log r_t|⁠ (default 1e-10).
split	Logical: split the posterior draws in half so the proposal is fit on one half and scored on the other (default TRUE, recommended). Set FALSE only for diagnostic comparison.
verbose	Print iteration history (default FALSE).

Value

A list with:

• log_marginal: the bridge-sampling estimate of \log Z.

• n_iter: bridge iterations to convergence.

• converged: logical.

• re_sd: relative MSE estimate (Fruehwirth-Schnatter 2004); a rough quality check, smaller is better.

• proposal: the fitted proposal list(mean, cov).

Tier

Bridge sampling is a post-hoc marginal-likelihood estimator, not a sampling backend, so it does not appear in the INFERENCE_TIERS registry. It operates on draws from any Tier-1 (Exact) backend.

References

Meng, X.-L., & Wong, W. H. (1996). Simulating ratios of normalizing constants via a simple identity: a theoretical exploration. Statistica Sinica, 6, 831-860.

Gronau, Q. F., Sarafoglou, A., Matzke, D., Ly, A., Boehm, U., Marsman, M., ... & Steingroever, H. (2017). A tutorial on bridge sampling. Journal of Mathematical Psychology, 81, 80-97.

Examples

# Toy: marginal likelihood of N(theta | 0, 1) under Y = theta + eps,
# eps ~ N(0, 1), prior theta ~ N(0, 10). Closed-form available.
y <- 1.5
log_post <- function(theta) {
  dnorm(y, theta, 1, log = TRUE) + dnorm(theta, 0, 10, log = TRUE)
}
draws <- matrix(rnorm(2000, mean = y * 100 / 101, sd = sqrt(100 / 101)),
                ncol = 1)
bs <- bridge_sampling(draws, log_post)
bs$log_marginal  # should match dnorm(y, 0, sqrt(101), log = TRUE)

tulpa documentation built on Oct. 19, 2026, 5:09 p.m.