A Probabilistic View on Predictive Constructions for Bayesian Learning

Given a sequence X=(X1,X2,…) of random observations, Bayesian forecaster aims to predict Xn+1 based on (X1,…,Xn) for each n≥0. To this end, in principle, she only needs select collection σ=(σ0,σ1,…), called “strategy” what follows, where σ0(·)=P(X1∈·) is the marginal distribution X1 and σn(·)=P(Xn+1∈·|X1,…,Xn) nth predictive distribution. Because Ionescu–Tulcea theorem, σ can be assigned directly, without passing through usual prior/posterior scheme. One main advantage that no prior probability selected. In nutshell, approach learning. A concise review latter provided paper. We try put such an right framework, make clear few misunderstandings, provide unifying view. Some recent results are discussed as well. addition, some new strategies introduced corresponding data X determined. The concern generalized Pólya urns, change points, covariates stationary sequences.