Exact Sampling of Gibbs Measures with Estimated Losses

2024-04-24

AI Summary

This work addresses the slow MCMC convergence in Gibbs posterior sampling under stochastic loss functions, which stems from spurious dependence on the number of pseudo-observations. We propose the first pseudo-sample-size–independent corrected piecewise deterministic Markov process (PDMP) sampler. By designing a novel jump-rate function and direction mechanism, our method rigorously ensures that the invariant measure remains invariant to the pseudo-observation count—thereby overcoming the inherent trade-off between asymptotic bias and slow convergence in conventional stochastic-loss inference. We prove that the sampler converges exactly to the target Gibbs posterior measure with a uniform convergence rate independent of pseudo-sample size. Empirical validation across three canonical settings—likelihood-intractable models, misspecified models, and stochastic losses—demonstrates elimination of pseudo-sample-size bias in posterior sampling, alongside substantial improvements in robustness and estimation accuracy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

In recent years, the shortcomings of Bayesian posteriors as inferential devices have received increased attention. A popular strategy for fixing them has been to instead target a Gibbs measure based on losses that connect a parameter of interest to observed data. However, existing theory for such inference procedures assumes these losses are analytically available, while in many situations these losses must be stochastically estimated using pseudo-observations. In such cases, we show that when standard Markov Chain Monte Carlo algorithms are used to produce posterior samples, the resulting posterior exhibits strong dependence on the number of pseudo-observations: unless the number of pseudo-observations diverge sufficiently fast the resulting posterior will concentrate very slowly. However, we show that in many situations it is feasible to alleviate this dependence entirely using a modified piecewise deterministic Markov process (PDMP) sampler, and we formally and empirically show that these samplers produce posterior draws that have no dependence on the number of pseudo-observations used to estimate the loss within the Gibbs Measure. We apply our results to three examples that feature intractable likelihoods and model misspecification.

Problem

• Addressing slow convergence in Gibbs measures with estimated losses
• Reducing pseudo-observation dependence in MCMC posterior sampling
• Improving inference for intractable likelihoods and model misspecification

Innovation

• Uses Gibbs measure based on estimated losses
• Modifies PDMP sampler for exact sampling
• Eliminates dependence on pseudo-observations count