Bayesian predictive inference without a prior

Let $(X_n:nge 1)$ be a sequence of random observations. Let $sigma_n(cdot)=Pigl(X_{n+1}in~cdotmid X_1,ldots,X_nigr)$ be the $n$-th predictive distribution and $sigma_0(cdot)$=$P(X_1in~cdot)$ the marginal distribution of $X_1$. To make predictions on $(X_n)$, a Bayesian forecaster only needs the collection $sigma=(sigma_n:nge 0)$. 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. This point of view is adopted in this paper. The choice of $sigma$ is only subjected to two requirements: (i) The resulting sequence $(X_n)$ is conditionally identically distributed, in the sense of cite{BPR2004}; (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 $n ightarrowinfty$, is determined. In some cases, the probability distribution of $(X_n)$ is also evaluated.