Bayesian, frequentist and fiducial intervals for the difference between two binomial proportions

TL;DR

This paper compares Bayesian, frequentist, and fiducial methods for estimating the difference between two binomial proportions, focusing on coverage probabilities and interval lengths to evaluate their performance.

Contribution

It introduces a comprehensive comparison of three types of priors and fiducial inference for binomial differences, highlighting their relative effectiveness.

Findings

Probability matching and divergence priors outperform Jeffreys prior.

Fiducial methods show comparable performance to Bayesian and classical methods.

Coverage rates and interval lengths are key metrics for evaluation.

Abstract

Estimating the difference between two binomial proportions will be investigated, where Bayesian, frequentist and fiducial (BFF) methods will be considered. Three vague priors will be used, the Jeffreys prior, a divergence prior and the probability matching prior. A probability matching prior is a prior distribution under which the posterior probabilities of certain regions coincide with their coverage probabilities. Fiducial inference can be viewed as a procedure that obtains a measure on a parameter space while assuming less than what Bayesian inference does, i.e. no prior. Fisher introduced the idea of fiducial probability and fiducial inference. In some cases the fiducial distribution is equivalent to the Jeffreys posterior. The performance of the Jeffreys prior, divergence prior and the probability matching prior will be compared to a fiducial method and other classical methods of constructing confidence intervals for the difference between two independent binomial parameters. These intervals will be compared and evaluated by looking at their coverage rates and average interval lengths. The probability matching and divergence priors perform better than the Jeffreys prior.