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February 2021 A class of models for Bayesian predictive inference
Patrizia Berti, Emanuela Dreassi, Luca Pratelli, Pietro Rigo
Bernoulli 27(1): 702-726 (February 2021). DOI: 10.3150/20-BEJ1255


In a Bayesian framework, to make predictions on a sequence $X_{1},X_{2},\ldots $ of random observations, the inferrer needs to assign the predictive distributions $\sigma _{n}(\cdot )=P(X_{n+1}\in \cdot \mid X_{1},\ldots,X_{n})$. In this paper, we propose to assign $\sigma _{n}$ directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence $(X_{n})$ is assumed to be conditionally identically distributed (c.i.d.) in the sense of (Ann. Probab. 32 (2004) 2029–2052). To realize this programme, a class $\Sigma $ of predictive distributions is introduced and investigated. Such a $\Sigma $ is rich enough to model various real situations and $(X_{n})$ is actually c.i.d. if $\sigma _{n}$ belongs to $\Sigma $. Furthermore, when a new observation $X_{n+1}$ becomes available, $\sigma _{n+1}$ can be obtained by a simple recursive update of $\sigma _{n}$. If $\mu $ is the a.s. weak limit of $\sigma _{n}$, conditions for $\mu $ to be a.s. discrete are provided as well.


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Patrizia Berti. Emanuela Dreassi. Luca Pratelli. Pietro Rigo. "A class of models for Bayesian predictive inference." Bernoulli 27 (1) 702 - 726, February 2021.


Received: 1 March 2020; Revised: 1 June 2020; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282867
MathSciNet: MR4177386
Digital Object Identifier: 10.3150/20-BEJ1255

Keywords: Bayesian nonparametrics , conditional identity in distribution , exchangeability , predictive distribution , random probability measure , sequential predictions , strategy

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability


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Vol.27 • No. 1 • February 2021
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