Efficient Bayes Factor Sensitivity Analysis via Posterior Density Ratios

2026-04-23

AI Summary

This work addresses the high sensitivity of Bayes factors to prior hyperparameters and the prohibitive computational cost of traditional sensitivity analyses, which require repeated model fits. The authors propose an efficient approach that combines anchor Bayes factors with the Savage–Dickey density ratio, enabling reconstruction of the full sensitivity curve with only a single additional model fit. They further introduce Importance-Weighted Marginal Density Estimation (IWMDE), which leverages existing MCMC samples to compute prior density ratios directly, thereby avoiding repeated likelihood evaluations. The method supports joint sensitivity analysis across multiple hyperparameters and facilitates Bayesian model averaging. Demonstrated on t-tests and meta-analyses, it yields highly accurate sensitivity curves at minimal computational cost, substantially outperforming existing techniques such as kernel density estimation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Bayes factor sensitivity analysis examines how the evidence for one hypothesis over another depends on the prior distribution. In complex models, the standard approach refits the model at each hyper-parameter value, and the total computational cost scales linearly in the grid size. We propose a method that recovers the entire sensitivity curve from a single additional model fit. The key identity decomposes the Bayes factor at any hyper-parameter value $γ_x$ into an ``anchor'' Bayes factor at a fixed reference $γ_0$ and a Savage--Dickey density ratio in an extended model that places a hyper-prior on $γ$. Once this extended model is fit, the Bayes factor at any $γ_x$ follows from the anchor value and a ratio of two posterior density ordinates. To approximate this ratio, we employ the importance-weighted marginal density estimator (IWMDE). Because the sensitivity parameter enters the model only through the prior distribution on the model parameters, the data likelihood cancels in the IWMDE, reducing it to a simple ratio of prior density evaluations on the MCMC draws, without any additional likelihood computation. The resulting estimator is fast, remains accurate even with small MCMC samples, and substantially outperforms kernel density estimation across the full sensitivity range. The method extends naturally to simultaneous sensitivity over multiple hyper-parameters and to Bayesian model averaging. We illustrate it on a univariate Bayesian $t$-test with exact Bayes factors for validation, a bivariate informed $t$-test, and a Bayesian model-averaged meta-analysis, obtaining accurate sensitivity curves at a fraction of the brute-force cost.

Problem

• Bayes factor
• sensitivity analysis
• computational efficiency
• posterior density ratios
• hyper-parameter

Innovation

• Bayes factor sensitivity analysis
• Savage–Dickey density ratio
• importance-weighted marginal density estimator
• posterior density ratios
• Bayesian model averaging