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

Abstract

Interval estimation is one of the most frequently used methods in statistical science, employed to provide a range of credible values a parameter is located in after taking into account the uncertainty in the data. However, while this interpretation only holds for Bayesian interval estimates, these suffer from two problems. First, Bayesian interval estimates can include values which have not been corroborated by observing the data. Second, Bayesian interval estimates and hypothesis tests can yield contradictory conclusions. In this paper a new theory for Bayesian hypothesis testing and interval estimation is presented. A new interval estimate is proposed, the Bayesian evidence interval, which is inspired by the Pereira-Stern theory of the full Bayesian significance test (FBST). It is shown that the evidence interval is a generalization of existing Bayesian interval estimates, that it solves the problems of standard Bayesian interval estimates and that it unifies Bayesian hypothesis testing and parameter estimation. The Bayesian evidence value is introduced, which quantifies the evidence for the (interval) null and alternative hypothesis. Based on the evidence interval and the evidence value, the (full) Bayesian evidence test (FBET) is proposed as a new, model-independent Bayesian hypothesis test. Additionally, a decision rule for hypothesis testing is derived which shows the relationship to a widely used decision rule based on the region of practical equivalence and Bayesian highest posterior density intervals and to the e-value in the FBST. In summary, the proposed method is universally applicable, computationally efficient, and while the evidence interval can be seen as an extension of existing Bayesian interval estimates, the FBET is a generalization of the FBST and contains it as a special case. Together, the theory developed provides a unification of Bayesian hypothesis testing and interval estimation and is made available in the R package fbst.