A Bayes Factor Framework for Unified Parameter Estimation and Hypothesis Testing

2024-03-14

AI Summary

This paper addresses key limitations of traditional Bayesian and frequentist inference frameworks—namely, their conceptual incompatibility, strong dependence on prior specification, and difficulty handling nuisance parameters. To resolve these issues, we propose a unified statistical inference framework grounded in the Bayes factor. Our core method treats the Bayes factor as a function of the null-hypothesis parameter value, enabling construction of a “support curve”; from this curve, we derive the maximum evidence estimator (a point estimate) and the support interval (an interval estimate), thereby integrating hypothesis testing and parameter inference. Key contributions include: (i) the first direct use of the Bayes factor for parameter estimation; (ii) introduction of novel concepts—support curve, maximum evidence estimator, and support interval; and (iii) elimination of prior specification and automatic resolution of nuisance-parameter issues, thus bridging the Bayesian–frequentist divide. Empirical applications in meta-analysis, replication studies, and logistic regression demonstrate improved interpretability, transparency, and reproducibility of statistical evidence.

Technology Category

Machine Learning: Bayesian LearningReasoning under Uncertainty: Probabilistic InferenceCognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systemsSearch and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Unify parameter estimation and hypothesis testing using Bayes factors
• Develop support curves for point and interval estimates inversion
• Provide practical framework for meta-analysis and regression applications

Innovation

• Bayes factor for unified hypothesis testing and estimation
• Support curve inversion for point and interval estimates
• Integrated likelihood ratios handling nuisance parameters