A Bayes Factor Framework for Unified Parameter Estimation and Hypothesis Testing

TL;DR

This paper introduces a unified framework using Bayes factors for both parameter estimation and hypothesis testing, providing a practical and visual approach for statistical inference without relying on prior synthesis.

Contribution

It presents a novel method to invert Bayes factors into support curves, enabling point and interval estimates similar to P-value functions, bridging Bayesian and frequentist inference.

Findings

Support curves facilitate intuitive inference visualization

Support intervals can be derived at any confidence level

Framework applied successfully to meta-analysis and logistic regression

Abstract

The Bayes factor, the data-based updating factor of the prior to posterior odds of two hypotheses, is a natural measure of statistical evidence for one hypothesis over the other. We show how Bayes factors can also be used for parameter estimation. The key idea is to consider the Bayes factor as a function of the parameter value under the null hypothesis. This `support curve' is inverted to obtain point estimates (`maximum evidence estimates') and interval estimates (`support intervals'), similar to how P-value functions are inverted to obtain point estimates and confidence intervals. This provides data analysts with a unified inference framework as Bayes factors (for any tested parameter value), support intervals (at any level), and point estimates can be easily read off from a plot of the support curve. This approach shares similarities but is also distinct from conventional Bayesian and frequentist approaches: It uses the Bayesian evidence calculus, but without synthesizing data and prior, and it defines statistical evidence in terms of (integrated) likelihood ratios, but also includes a natural way for dealing with nuisance parameters. Applications to meta-analysis, replication studies, and logistic regression illustrate how our framework is of practical value for making quantitative inferences.