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In this paper, we start by reviewing exchangeability and its relevance to the Bayesian approach. We highlight the predictive nature of Bayesian models and the symmetry assumptions implied by beliefs of an underlying exchangeable sequence of observations. By taking a closer look at the Bayesian bootstrap, the parametric bootstrap of Efron and a version of Bayesian thinking about inference uncovered by Doob based on martingales, we introduce a parametric Bayesian bootstrap. Martingales play a fundamental role. Illustrations are presented as is the relevant theory. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Original publication

DOI

10.1098/rsta.2022.0143

Type

Journal article

Journal

Philos Trans A Math Phys Eng Sci

Publication Date

15/05/2023

Volume

381

Keywords

bootstrap, parametric bootstrap, predictive inference, score function