A class of models for Bayesian predictive inference

In a Bayesian framework, to make predictions on a sequence $X_1,X_2,ldots$ of random observations, the inferrer needs to assign the predictive distributions $sigma_n(cdot)=Pigl(X_{n+1}incdotmid X_1,ldots,X_nigr)$. In this paper, we propose to assign $sigma_n$ directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence $(X_n)$ is assumed to be conditionally identically distributed (c.i.d.) in the sense of cite{BPR2004}. To realize this programme, a class $Sigma$ of predictive distributions is introduced and investigated. Such a $Sigma$ is rich enough to model various real situations and $(X_n)$ is actually c.i.d. if $sigma_n$ belongs to $Sigma$. Furthermore, when a new observation $X_{n+1}$ becomes available, $sigma_{n+1}$ can be obtained by a simple recursive update of $sigma_n$. If $mu$ is the a.s. weak limit of $sigma_n$, conditions for $mu$ to be a.s. discrete are provided as well.