2023 A Probabilistic View on Predictive Constructions for Bayesian Learning
Patrizia Berti, Emanuela Dreassi, Fabrizio Leisen, Luca Pratelli, Pietro Rigo
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Statist. Sci. Advance Publication 1-15 (2023). DOI: 10.1214/23-STS884

Abstract

Given a sequence X=(X1,X2,) of random observations, a Bayesian forecaster aims to predict Xn+1 based on (X1,,Xn) for each n0. 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.

Acknowledgments

We are grateful to Federico Bassetti and Paola Bortot for very useful conversations.

Citation

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Patrizia Berti. Emanuela Dreassi. Fabrizio Leisen. Luca Pratelli. Pietro Rigo. "A Probabilistic View on Predictive Constructions for Bayesian Learning." Statist. Sci. Advance Publication 1 - 15, 2023. https://doi.org/10.1214/23-STS884

Information

Published: 2023
First available in Project Euclid: 23 February 2023

Digital Object Identifier: 10.1214/23-STS884

Keywords: Bayesian inference , conditional identity in distribution , exchangeability , predictive distribution , sequential predictions , stationarity

Rights: Copyright © 2023 Institute of Mathematical Statistics

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