A probabilistic view on predictive constructions for Bayesian learning

Given a sequence X = (X1, X2, . . .) of random observations, a Bayesian forecaster aims to predict Xn+1 based on (X1, . . . , Xn) for each n ≥ 0. To this end, in principle, she only needs to select a collection σ = (σ0,σ1, . . .), called “strategy” in what follows, where σ0(·) = P(X1 ∈ ·) is the marginal distribution of X1 and σn(·) = P(Xn+1 ∈ ·|X1, . . . , Xn) the nth 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 Pólya urns, random change points, covariates and stationary sequences.