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. In a Bayesian framework, to make predictions on (Xn), one 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. In this paper, σ is subjected to two requirements: (i) The resulting sequence (Xn) is conditionally identically distributed, in the sense of Berti, Pratelli and Rigo (2004); (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.