Bayesian predictive inference without a prior

Let (Xn:n≥ 1) be a sequence of random observations. Let σn(·)=Pbigl(Xn+1∈·mid X1,ldots,Xnbigr) 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 nrightarrow∞, is determined. In some cases, the probability distribution of (Xn) is also evaluated.