A predictive approach to Bayesian forecasting
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Given a sequence X=(X_1,X_2,…) of random observations, a Bayesian forecaster aims to predict X_n+1 based on (X_1,…,X_n) for each n≥ 0. To this end, she only needs to select a collection σ=(σ_0,σ_1,…), called "strategy" in what follows, where σ_0(·)=P(X_1∈·) is the marginal distribution of X_1 and σ_n(·)=P(X_n+1∈·| X_1,…,X_n) 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 non-standard approach to Bayesian predictive inference. 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 Polya urns, random change points, covariates and stationary sequences.
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