Pietro Rigo's contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker
本文提出以缺失观测而非先验分布作为贝叶斯不确定性来源的新视角，通过鞅后验分布直接指定联合预测，适用于密度估计、回归和分类。
The prior distribution is the usual starting point for Bayesian uncertainty.In this paper, we present a different perspective that focuses on missing observations as the source of statistical uncertainty, with the parameter of interest being known precisely given the entire population.We argue that the foundation of Bayesian inference is to assign a distribution on missing observations conditional on what has been observed.In the i.i.d.setting with an observed sample of size n, the Bayesian would thus assign a predictive distribution on the missing Y n+1: conditional on Y 1:n , which then induces a distribution on the parameter.We utilize Doob's theorem, which relies on martingales, to show that choosing the Bayesian predictive distribution returns the conventional posterior as the distribution of the parameter.Taking this as our cue, we relax the predictive machine, avoiding the need for the predictive to be derived solely from the usual prior to posterior to predictive density formula.We introduce the martingale posterior distribution, which returns Bayesian uncertainty on any statistic via the direct specification of the joint predictive.To that end, we introduce new predictive methodologies for multivariate density estimation, regression and classification that build upon recent work on bivariate copulas.