Promise of Bayesian Inference for Astrophysics

Abstract

The “frequentist” approach to statistics, currently dominating statistical practice in astrophysics, is compared to the historically older Bayesian approach, which is now growing in popularity in other scientific disciplines, and which provides unique, optimal solutions to well-posed problems. The two approaches address the same questions with very different calculations, but in simple cases often give the same final results, confusing the issue of whether one is superior to the other. Here frequentist and Bayesian methods are applied to problems where such a mathematical coincidence does not occur, allowing assessment of their relative merits based on their performance, rather than philosophical argument. Emphasis is placed on a key distinction between the two approaches: Bayesian methods, based on comparisons among alternative hypotheses using the single observed data set, consider averages over hypotheses; frequentist methods, in contrast, average over hypothetical alternative data samples and consider hypothesis averaging to be irrelevant. Simple problems are presented that magnify the consequences of this distinction to where common sense can confidently judge between the methods. These demonstrate the irrelevance of sample averaging, and the necessity of hypothesis averaging, revealing frequentist methods to be fundamentally flawed. Bayesian methods are then presented for astrophysically relevant problems using the Poisson distribution, including the analysis of “on/off” measurements of a weak source in a strong background. Weaknesses of the presently used frequentist methods for these problems are straightforwardly overcome using Bayesian methods. Additional existing applications of Bayesian inference to astrophysical problems are noted.