Bayesian Predictive Inference Without a Prior

Abstract

Let (Xn : n ≥ 1) be a sequence of random observations. Let σn(•) = P Xn+1 ∈ • | X1, . . . , Xn be the n-th predictive distribution and σ0(•)=P (X1 ∈ •) the marginal distribution of X1. To make predictions on (Xn), a Bayesian forecaster only needs the collection σ = (σn : n ≥ 0). Because of the Ionescu-Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be selected. This point of view is adopted in this paper. The choice of σ is only subjected to two requirements: (i) The resulting sequence (Xn) is conditionally identically distributed, in the sense of [4]; (ii) Each σn+1 is a simple recursive update of σn. Various new σ satisfying (i)-(ii) are introduced and investigated.For such σ, the asymptotics of σn, as n → ∞, is determined. In some cases, the probability distribution of (Xn) is also evaluated.