A class of models for Bayesian predictive inference

Berti, Patrizia; Dreassi, Emanuela; Pratelli, Luca; Rigo, Pietro

doi:10.3150/20-bej1255

Cited by 13 publications

(34 citation statements)

References 22 publications

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“…For some reason, we would like to replace (U n , V n ) with another sequence X n = (Y , Z n ), possibly defined on a different probability space, such that X n a.s. −→ X 0 and X n ∼ (U n , V n ) for all n. Note that all the X n have the same first coordinate Y , and this is basic in various frameworks, including optimal transport and stochastic control. Corollary 2 states that, under mild conditions, to replace (U n , V n ) with X n is admissible if and only if condition (6) holds with μ n the probability distribution of (U n , V n ). See also [4,Prop.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…An exchangeable sequence is c.i.d. but not conversely; see, e.g., [6] and references therein. However, as in the exchangeable case, if Y is c.i.d.…”

Section: Examplesmentioning

confidence: 99%

“…We do not know whether this condition holds for any c.i.d. sequence, but it holds in some (meaningful) special cases, including N a.s. discrete; see also [6,Th. 18].…”

Section: Examplesmentioning

confidence: 99%

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A Strong Version of the Skorohod Representation Theorem

Pratelli¹,

Rigo

2022

J Theor Probab

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For each $$n\ge 0$$ n ≥ 0 , let $$\mu _n$$ μ n be a tight probability measure on the Borel $$\sigma $$ σ -field of a metric space S. Let $$(T,{\mathcal {C}})$$ ( T , C ) be a measurable space such that the diagonal $$\bigl \{(t,t):t\in T\bigr \}$$ { ( t , t ) : t ∈ T } belongs to $${\mathcal {C}}\otimes {\mathcal {C}}$$ C ⊗ C . Fix a measurable function $$g:S\rightarrow T$$ g : S → T and suppose $$\mu _n=\mu _0$$ μ n = μ 0 on $$g^{-1}({\mathcal {C}})$$ g - 1 ( C ) for all $$n\ge 0$$ n ≥ 0 . Necessary and sufficient conditions for the existence of S-valued random variables $$X_n$$ X n , defined on the same probability space and satisfying $$\begin{aligned} X_n\overset{\text {a.s.}}{\longrightarrow }X_0,\quad X_n\sim \mu _n\,\text { and } \,g(X_n)=g(X_0)\,\text { for all }n\ge 0, \end{aligned}$$ X n ⟶ a.s. X 0 , X n ∼ μ n and g ( X n ) = g ( X 0 ) for all n ≥ 0 , are given. Such conditions are then applied to several examples. The tightness condition on $$\mu _0$$ μ 0 can be dropped at the price of some assumptions on S and g.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…An exchangeable sequence is c.i.d. but not conversely; see, e.g., [6] and references therein. However, as in the exchangeable case, if Y is c.i.d.…”

Section: Examplesmentioning

confidence: 99%

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A Strong Version of the Skorohod Representation Theorem

Pratelli¹,

Rigo

2022

J Theor Probab

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“…The resulting measure-valued urn process provides a predictive characterization of the law of an asymptotically exchangeable sequence of random variables, which corresponds to the observation process of an implied urn sampling scheme. In fact, the model ( 6)-( 7) fits into a line of recent research, which explores efficient predictive constructions for fast online prediction or approximately-Bayesian solutions, see [11,29,32] and references therein. To that end, one direction for future work is to generalize the functional relationship in (7) and/or, as one referee suggested, to consider finitely-additive measures, along the lines discussed in [33].…”

Section: Discussionmentioning

confidence: 99%

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

2021

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Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

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“…For definiteness, as in [8], we call them the standard approach (SA) and the non-standard approach (NSA). Both are admissible, from the Bayesian point of view, and both lead to a full specification of the probability distribution of (X n ).…”

Section: Motivationsmentioning

confidence: 99%

Bayesian Predictive Inference Without a Prior

Berti¹,

Dreassi²,

Leisen³

et al. 2024

STAT SINICA

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Let (Xn : n ≥ 1) be a sequence of random observations. Let σn(•) = P Xn+1 ∈ • | X1, . . . , Xn be the n-th predictive distribution and σ0(•)=P (X1 ∈ •) the marginal distribution of X1. To make predictions on (Xn), a Bayesian forecaster only needs the collection σ = (σn : n ≥ 0). 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 has to be selected. This point of view is adopted in this paper. The choice of σ is only subjected to two requirements: (i) The resulting sequence (Xn) is conditionally identically distributed, in the sense of [4]; (ii) Each σn+1 is a simple recursive update of σn. Various new σ satisfying (i)-(ii) are introduced and investigated.For such σ, the asymptotics of σn, as n → ∞, is determined. In some cases, the probability distribution of (Xn) is also evaluated.

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A class of models for Bayesian predictive inference

Cited by 13 publications

References 22 publications

A Strong Version of the Skorohod Representation Theorem

A Strong Version of the Skorohod Representation Theorem

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Bayesian Predictive Inference Without a Prior

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