Statistical inference with exchangeability and martingales

Abstract

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'.

Keywords: bootstrap; parametric bootstrap; predictive inference; score function.

Figure 1.

Marginal frequentist posterior distribution for $\theta$ and ${\sigma }^2$.

Figure 2.

Frequentist posterior (histogram) from the Poisson example, alongside the Bayesian gamma objective posterior (line).

Figure 3.

(a and b) Frequentist posteriors from the gamma example.

Figure 4.

Frequentist posteriors from the linear model example.

Figure 5.

Frequentist posterior from the Markov time series model example.