Overlap, Not Moments: Finite-Sample Theory for Bridge Sampling
Abstract
Bridge sampling estimates of marginal likelihoods are routinely reported with a delta-method Monte Carlo standard error (MCSE). Existing theory is asymptotic and cannot say whether a given run is long enough to trust the estimate or its MCSE. We give finite-sample answers. The optimal bridge estimator fits a logistic regression that separates posterior from proposal draws, and its accuracy is governed by their overlap, within a factor of two of the classification's Bayes error. Because the estimating function is a sum of bounded terms, no moment conditions are needed, and all bounds hold uniformly over targets, proposals and dimensions. A number of draws inversely proportional to the overlap suffices and is necessary in the worst case, paralleling the exponential-in-Kullback--Leibler requirement of importance sampling. A Berry--Esseen bound with an explicit constant holds at the optimal rate, and the MCSE is accurate once the effective overlap is large; it is estimated at no cost. For Markov chain draws, the required sample size grows in proportion to the sampler's relaxation time, unavoidably in the worst case. The estimated bridge, the fixed-point iteration and the fitted proposal matter only at second order, and we give free diagnostics for sampler failure and misestimated autocorrelation.
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BibTeX
@misc{micaletto2026overlap,
title = {{Overlap, Not Moments: Finite-Sample Theory for Bridge Sampling}},
author = {Giorgio Micaletto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17403},
eprint = {2610.17403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17403},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Overlap, Not Moments: Finite-Sample Theory for Bridge Sampling
AU - Giorgio Micaletto
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17403
UR - https://arxiv.org/abs/2610.17403
AB - Bridge sampling estimates of marginal likelihoods are routinely reported with a delta-method Monte Carlo standard error (MCSE). Existing theory is asymptotic and cannot say whether a given run is long enough to trust the estimate or its MCSE. We give finite-sample answers. The optimal bridge estimator fits a logistic regression that separates posterior from proposal draws, and its accuracy is governed by their overlap, within a factor of two of the classification's Bayes error. Because the estimating function is a sum of bounded terms, no moment conditions are needed, and all bounds hold uniformly over targets, proposals and dimensions. A number of draws inversely proportional to the overlap suffices and is necessary in the worst case, paralleling the exponential-in-Kullback--Leibler requirement of importance sampling. A Berry--Esseen bound with an explicit constant holds at the optimal rate, and the MCSE is accurate once the effective overlap is large; it is estimated at no cost. For Markov chain draws, the required sample size grows in proportion to the sampler's relaxation time, unavoidably in the worst case. The estimated bridge, the fixed-point iteration and the fitted proposal matter only at second order, and we give free diagnostics for sampler failure and misestimated autocorrelation.
ER -