Bayesian predictive inference without a prior

TL;DR

This paper introduces a prior-free Bayesian predictive framework where the sequence of predictive distributions is directly specified and updated recursively, ensuring conditional identical distribution and analyzing their asymptotic behavior.

Contribution

It proposes a novel approach to Bayesian prediction that bypasses priors by directly defining and updating predictive distributions with proven asymptotic properties.

Findings

New classes of predictive distributions satisfying the recursive and conditional i.i.d. properties.

Asymptotic behavior of these predictive distributions is characterized.

Distributional properties of the observed sequence are derived in some cases.

Abstract

Let $(X_n:n\ge 1)$ be a sequence of random observations. Let $\sigma_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the $n$-th predictive distribution and $\sigma_0(\cdot)=P(X_1\in\cdot)$ the marginal distribution of $X_1$. In a Bayesian framework, to make predictions on $(X_n)$, one only needs the collection $\sigma=(\sigma_n:n\ge 0)$. Because of the Ionescu-Tulcea theorem, $\sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be selected. In this paper, $\sigma$ is subjected to two requirements: (i) The resulting sequence $(X_n)$ is conditionally identically distributed, in the sense of Berti, Pratelli and Rigo (2004); (ii) Each $\sigma_{n+1}$ is a simple recursive update of $\sigma_n$. Various new $\sigma$ satisfying (i)-(ii) are introduced and investigated. For such $\sigma$, the asymptotics of $\sigma_n$, as $n\rightarrow\infty$, is determined. In some cases, the probability distribution of $(X_n)$ is also evaluated.