Bayesian, classical and hybrid methods of inference when one parameter value is special

TL;DR

This paper explores Bayesian, classical, and hybrid inference methods for a parameter when a specific interval around a value is considered special, emphasizing prior belief and practical applications in medical data analysis.

Contribution

It introduces a pseudo-Bayesian approach based on sensitivity analysis and proposes hybrid inference methods that do not strictly rely on Bayesian probabilities.

Findings

Pseudo-Bayesian method based on lower limits of posterior probability.

Hybrid methods combining classical significance and Bayesian ideas.

Application to real medical meta-analysis data.

Abstract

This paper considers the problem of making statistical inferences about a parameter when a narrow interval centred at a given value of the parameter is considered special, which is interpreted as meaning that there is a substantial degree of prior belief that the true value of the parameter lies in this interval. A clear justification of the practical importance of this problem is provided. The main difficulty with the standard Bayesian solution to this problem is discussed and, as a result, a pseudo-Bayesian solution is put forward based on determining lower limits for the posterior probability of the parameter lying in the special interval by means of a sensitivity analysis. Since it is not assumed that prior beliefs necessarily need to be expressed in terms of prior probabilities, nor that post-data probabilities must be Bayesian posterior probabilities, hybrid methods of inference are also proposed that are based on specific ways of measuring and interpreting the classical concept of significance. The various methods that are outlined are compared and contrasted at both a foundational level, and from a practical viewpoint by applying them to real data from meta-analyses that appeared in a well-known medical article.