Positioning Bayesian inference as a particular application of frequentist inference and vice versa

Abstract

Bayesian and frequentist inference are sometimes positioned as two mutually exclusive alternatives for analysing data and drawing conclusions. Here, a framework is proposed to position Bayesian inference as a particular application of frequentist inference and vice versa. As a result Bayesian inference and frequentist inference can be seen as two special cases of the more general problem of predicting future observations based on existing observations. In this light Bayesian and frequentist inference appear as two complementary and compatible approaches to relate data to probability models. The two approaches ask different questions. For a given problem, one may decide which of the questions to ask and whether to use any one of the two or combine both approaches.