A Probabilistic View on Predictive Constructions for Bayesian Learning

Patrizia Berti

doi:10.1214/23-STS884

A Probabilistic View on Predictive Constructions for Bayesian Learning

2025-01-01

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DOI：10.1214/23-STS884

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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 6 = (60, 61, ...), called strategy in what follows, where 60() = P(X1 is an element of ) is the marginal distribution of X1 and 6n() = P (Xn+1 is an element of |X1, ... , Xn) the nth predictive distribution. Because of the Ionescu-Tulcea theorem, 6 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 & oacute;lya urns, random change points, covariates and stationary sequences.

Keywords:

Bayesian inference

conditional identity in distri- bution

exchangeability

predictive distribution

sequential predictions

sta- tionarity

A Probabilistic View on Predictive Constructions for Bayesian Learning

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