A probabilistic view on predictive constructions for Bayesian learning

Given a sequence X=(X1,X2,ldots) of random observations, a Bayesian forecaster aims to predict Xn+1 based on (X1,ldots,Xn) for each n≥ 0. To this end, in principle, she only needs to select a collection σ=(σ0,σ1,ldots), called ``strategy" in what follows, where σ0(·)=P(X1∈·) is the marginal distribution of X1 and σn(·)=P(Xn+1∈·mid X1,ldots,Xn) the n-th predictive distribution. 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 is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized P'olya urns, random change points, covariates and stationary sequences.