Product Monte Carlo: Expectation Under a Product of Densities
Abstract
Aggregating independently computed sub-sample inferences is a well-known challenge in large-scale Bayesian computation, federated learning, and opinion pooling. The core problem is to approximate a distribution whose density is proportional to a product of individual densities. We propose Product Monte Carlo (PMC), an algorithm for computing arbitrary expectations under a product of densities using only Monte Carlo draws from individual densities. We turn the intractable product expectation into a regression problem by expressing it through a family of computable conditional expectations. Besides the point estimate, our fully Bayesian regression formulation provides coherent uncertainty quantification and principled diagnostics. We establish theoretical guarantees on consistency and characterize convergence rates for the PMC estimator. Across simulated and real-data experiments varying the number, dimensionality, and heterogeneity of the component distributions, PMC achieves higher estimation accuracy, better-calibrated uncertainty coverage, and improved scalability at a lower computational cost than existing methods. In addition, PMC requires no density evaluation, making it applicable to simulation-based inference, as we demonstrate in an application with protein images.
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BibTeX
@misc{carlotti2026product,
title = {{Product Monte Carlo: Expectation Under a Product of Densities}},
author = {Pietro Carlotti and Sifan Liu and Yuling Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12702},
eprint = {2610.12702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12702},
}
RIS
TY - UNPB
TI - Product Monte Carlo: Expectation Under a Product of Densities
AU - Pietro Carlotti
AU - Sifan Liu
AU - Yuling Yao
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12702
UR - https://arxiv.org/abs/2610.12702
AB - Aggregating independently computed sub-sample inferences is a well-known challenge in large-scale Bayesian computation, federated learning, and opinion pooling. The core problem is to approximate a distribution whose density is proportional to a product of individual densities. We propose Product Monte Carlo (PMC), an algorithm for computing arbitrary expectations under a product of densities using only Monte Carlo draws from individual densities. We turn the intractable product expectation into a regression problem by expressing it through a family of computable conditional expectations. Besides the point estimate, our fully Bayesian regression formulation provides coherent uncertainty quantification and principled diagnostics. We establish theoretical guarantees on consistency and characterize convergence rates for the PMC estimator. Across simulated and real-data experiments varying the number, dimensionality, and heterogeneity of the component distributions, PMC achieves higher estimation accuracy, better-calibrated uncertainty coverage, and improved scalability at a lower computational cost than existing methods. In addition, PMC requires no density evaluation, making it applicable to simulation-based inference, as we demonstrate in an application with protein images.
ER -