The evidence interval and the Bayesian evidence value: On a unified theory for Bayesian hypothesis testing and...

Area of Science:

• Statistical Science
• Bayesian Inference
• Hypothesis Testing

Background:

• Bayesian interval estimates are widely used but can include uncorroborated values and conflict with hypothesis tests.
• Existing Bayesian methods present challenges in reconciling parameter estimation with hypothesis testing.
• Standard Bayesian interval estimates may not fully reflect observed data or align with significance test outcomes.

Purpose of the Study:

• To present a new theory for Bayesian hypothesis testing and interval estimation.
• To introduce the Bayesian evidence interval and Bayesian evidence value.
• To unify Bayesian hypothesis testing and parameter estimation into a single framework.

Main Methods:

• Development of the Bayesian evidence interval, inspired by the full Bayesian significance test (FBST).
• Introduction of the Bayesian evidence value to quantify evidence for null and alternative hypotheses.
• Proposal of the full Bayesian evidence test (FBET) as a model-independent Bayesian hypothesis test.

Main Results:

• The Bayesian evidence interval is shown to be a generalization of existing Bayesian interval estimates.
• The proposed method solves problems associated with standard Bayesian interval estimates.
• The FBET is a generalization of the FBST, unifying hypothesis testing and interval estimation.

Conclusions:

• The new theory provides a universally applicable and computationally efficient method for Bayesian inference.
• The Bayesian evidence interval extends existing interval estimation techniques.
• The FBET generalizes the FBST, offering a unified approach to Bayesian hypothesis testing and parameter estimation.