Improving power posterior estimation of statistical evidence

TL;DR

This paper enhances the power posterior method for estimating Bayesian evidence by reducing bias through numerical analysis techniques and improving inverse temperature ladder selection, applicable also to Stepping Stone estimators.

Contribution

It introduces a numerical analysis-based bias reduction technique and a method for optimal inverse temperature ladder selection for power posterior evidence estimation.

Findings

Bias in evidence estimates can be significantly reduced.

The inverse temperature ladder can be optimized for better accuracy.

Method improves evidence estimation in Bayesian model comparison.

Abstract

The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation. This paper focuses on refining the power posterior approach to improve estimation of the evidence. The power posterior method involves transitioning from the prior to the posterior by powering the likelihood by an inverse temperature. In common with other tempering algorithms, the power posterior involves some degree of tuning. The main contributions of this article are twofold -- we present a result from the numerical analysis literature which can reduce the bias in the estimate of the evidence by addressing the error arising from numerically integrating across the inverse temperatures. We also tackle the selection of the inverse temperature ladder, applying this approach additionally to the Stepping Stone sampler estimation of evidence.