Extrapolation of Tempered Posteriors

2025-09-15

AI Summary

Sequential Monte Carlo (SMC) estimation of posterior expectations in Bayesian computation suffers from high variance and necessitates accurate approximation of intermediate distributions. Method: This paper proposes a novel posterior estimation framework based on tempered posterior extrapolation. Leveraging the key insight that the posterior expectation is uniquely determined by tempered expectations over any non-empty temperature interval, the method bypasses conventional stepwise annealing toward the target distribution. It constructs a sequence of intermediate distributions via temperature annealing, employs SMC sampling to obtain tempered expectations, and applies post-hoc smoothing and extrapolation to efficiently estimate the target posterior expectation. Contribution/Results: Theoretical analysis and empirical evaluation demonstrate that the proposed method substantially reduces estimation variance, improves computational efficiency, and exhibits strong numerical stability and generalization capability across diverse models and settings.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Tempering is a popular tool in Bayesian computation, being used to transform a posterior distribution $p_1$ into a reference distribution $p_0$ that is more easily approximated. Several algorithms exist that start by approximating $p_0$ and proceed through a sequence of intermediate distributions $p_t$ until an approximation to $p_1$ is obtained. Our contribution reveals that high-quality approximation of terms up to $p_1$ is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish conditions under which posterior expectations are determined by their associated tempered expectations on any non-empty $t$ interval. Harnessing this result, we propose novel methodology for approximating posterior expectations based on extrapolation and smoothing of tempered expectations, which we implement as a post-processing variance-reduction tool for sequential Monte Carlo.

Problem

• Extrapolating posterior expectations from tempered distributions
• Establishing conditions for posterior expectation determination
• Proposing variance-reduction via extrapolation and smoothing

Innovation

• Extrapolates posterior expectations from tempered distributions
• Uses non-empty interval tempered expectations
• Implements variance-reduction via smoothing extrapolation