Exchangeable sequences driven by an absolutely continuous random measure

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro

doi:10.1214/12-aop786

Cited by 18 publications

(18 citation statements)

References 7 publications

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“…The proof is based on a remarkable result in Berti et al . (), theorems 1 and 4, which shows that, for CID processes, the directing measure is absolutely continuous with respect to λ if and only if the predictive distribution is absolutely continuous and converges in total variation. This result does not apply directly to our setting, because ( θ n ) is not, generally, CID.…”

Section: A Statistical Interpretation Of Newton's Algorithmmentioning

confidence: 84%

“…In the on‐line appendix we provide two lemmas (lemmas A1.1 and A1.2), which are slight extensions of theorems 1 and 4 in Berti et al . (), the main difference being that we substitute the conditional identical distribution assumption with a martingale property that holds in our setting. These results lead to the following theorem.…”

Section: A Statistical Interpretation Of Newton's Algorithmmentioning

confidence: 99%

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Quasi-Bayes Properties of a Procedure for Sequential Learning in Mixture Models

Fortini

Petrone

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Bayesian methods are often optimal, yet increasing pressure for fast computations, especially with streaming data, brings renewed interest in faster, possibly suboptimal, solutions. The extent to which these algorithms approximate Bayesian solutions is a question of interest, but often unanswered. We propose a methodology to address this question in predictive settings, when the algorithm can be reinterpreted as a probabilistic predictive rule. We specifically develop the proposed methodology for a recursive procedure for on‐line learning in non‐parametric mixture models, which is often referred to as Newton's algorithm. This algorithm is simple and fast; however, its approximation properties are unclear. By reinterpreting it as a predictive rule, we can show that it underlies a statistical model which is, asymptotically, a Bayesian, exchangeable mixture model. In this sense, the recursive rule provides a quasi‐Bayes solution. Although the algorithm offers only a point estimate, our clean statistical formulation enables us to provide the asymptotic posterior distribution and asymptotic credible intervals for the mixing distribution. Moreover, it gives insights for tuning the parameters, as we illustrate in simulation studies, and paves the way to extensions in various directions. Beyond mixture models, our approach can be applied to other predictive algorithms.

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Section: A Statistical Interpretation Of Newton's Algorithmmentioning

confidence: 84%

Section: A Statistical Interpretation Of Newton's Algorithmmentioning

confidence: 99%

Quasi-Bayes Properties of a Procedure for Sequential Learning in Mixture Models

Fortini

Petrone

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…sequences have been introduced in [4] and [22] and then investigated in various papers; see e.g. [1], [2], [5], [6], [7], [8], [9], [11], [15], [18], [19].…”

Section: Conditional Identity In Distributionmentioning

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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“…sequences have been introduced in [4] and [22] and then investigated in various papers; see e.g. [1], [2], [5], [6], [7], [8], [9], [11], [14], [17], [18].…”

Section: Conditional Identity In Distributionmentioning

confidence: 99%

Bayesian predictive inference without a prior

Berti¹,

Dreassi²,

Leisen³

et al. 2021

Preprint

Self Cite

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Let (Xn : n ≥ 1) be a sequence of random observations. Let σn(•) = P X n+1 ∈ • | X 1 , . . . , Xn be the n-th predictive distribution and σ 0 (•) = P (X 1 ∈ •) the marginal distribution of X 1 . In a Bayesian framework, to make predictions on (Xn), one 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. In this paper, σ is subjected to two requirements: (i) The resulting sequence (Xn) is conditionally identically distributed, in the sense of [4]; (ii) Each σ n+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.

show abstract

Exchangeable sequences driven by an absolutely continuous random measure

Cited by 18 publications

References 7 publications

Quasi-Bayes Properties of a Procedure for Sequential Learning in Mixture Models

Quasi-Bayes Properties of a Procedure for Sequential Learning in Mixture Models

Bayesian Predictive Inference Without a Prior

Bayesian predictive inference without a prior

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