Martingale posterior distributions

Fong, Edwin; Holmes, Chris; Walker, Stephen G.

doi:10.48550/arxiv.2103.15671

Cited by 7 publications

(19 citation statements)

References 60 publications

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“…[23] proposed an approach combining generalised variational inference, PAC-Bayes and other approaches into a single principled framework; they give conditions for consistency of their approach but not for efficiency. [14] propose a novel view to constructing a generalised posterior distribution, so it would be interesting to study its efficiency.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

Bernstein - von Mises theorem and misspecified models: a review

Bochkina¹

2022

Preprint

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IntroductionConsider a family of probability models P (Y | θ) indexed by parameter θ ∈ Θ for observations y, and a prior distribution π on the parameter space Θ. In a classical Bayesian approach, the posterior distributionDenote the true distribution of observations P 0 , and we consider the casethe model is misspecified. Such case arises in many applications, particularly in complex models where the numerical evaluation of posterior distribution under the ideal probability model takes a long time to run, leading to increased use of approximate models with faster computing time. A typical example is approximating complex dependence structure by pairwise dependence only [2], [40].For well-specified regular models, the classical Bernstein -von Mises theorem states that for n independent identically distributed (iid) observations, for large n, the posterior distribution behaves approximately as a normal distribution centered on the true value of the parameter with a random shift, with both the posterior variance and the variance of the random shift being asymptotically equal to the inverse Fisher information, thus making the Bayesian inference asymptotically consistent and efficient.This was extended to locally asymptotically normal (LAN) models for Θ ∈ R p with fixed p [39], for LAN models with growing p, etc.A version of Bernstein -von Mises theorem is also available for nonregular models such as locally asymptotically exponential models ([20], [9]), models with parameter on the boundary of the parameter space [7] which hold under misspecified models. Here, however, we focus on "regular" models where estimators are asymptotically Gaussian.However, under model misspecification, Bayesian model is no longer optimal, as the posterior variance does not match the minimal lower bound on the variance of unbiased estimators [22] and [29]).Therefore, the standard way of constructing a posterior distribution following the Bayes theorem may not be appropriate for particular purposes, e.g. inference or prediction [27]. Different ways to construct a distribution of θ given Y that produces inference appropriate for the purpose of the analysis have been proposed. A natural aim for such a method is to behave like a standard Bayesian method when the

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Section: Discussion and Open Questionsmentioning

confidence: 99%

Bernstein - von Mises theorem and misspecified models: a review

Bochkina¹

2022

Preprint

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“…• A meaningful part of the usual Bayesian machinery can be developed under the sole assumption that (X n ) is c.i.d. ; see [14]. • A number of interesting strategies cannot be used if (X n ) is exchangeable, but are available if (X n ) is only required to be c.i.d.…”

Section: Motivationsmentioning

confidence: 99%

“…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%

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Bayesian predictive inference without a prior

Berti¹,

Dreassi²,

Leisen³

et al. 2021

Preprint

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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.

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“…The framing of causal analysis as a population missing data problem builds on the work of Fong et al (2021) who developed the idea for non-causal statistical inference, and Rubin (1978) who considered a generative model for potential outcomes under a dependent treatment assignment mechanism. A key difference between our proposal and the potential outcome framework in Rubin (1978) is that we condition on the observed data without any obligation to model what might have occurred for the observed units if things had been different.…”

Section: Introductionmentioning

confidence: 99%

Causal predictive inference and target trial emulation

Yiu¹,

Fong²,

Walker³

et al. 2022

Preprint

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Causal inference from observational data can be viewed as a missing data problem arising from a hypothetical population-scale randomized trial matched to the observational study. This links a target trial protocol with a corresponding generative predictive model for inference, providing a complete framework for transparent communication of causal assumptions and statistical uncertainty on treatment effects, without the need for counterfactuals. The intuitive foundation for the work is that a whole population randomized trial would provide answers to any observable causal question with certainty. Thus, our fundamental problem of causal inference is the missingness of the hypothetical target trial data, which we solve through repeated imputation from a generative predictive model conditioned on the observational data. Causal assumptions map to intuitive conditions on the transportability of predictive models across populations and conditions. We demonstrate our approach on a real data application to studying the effects of maternal smoking on birthweights using extensions of Bayesian additive regression trees and inverse probability weighting.

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Martingale posterior distributions

Cited by 7 publications

References 60 publications

Bernstein - von Mises theorem and misspecified models: a review

Bernstein - von Mises theorem and misspecified models: a review

Bayesian predictive inference without a prior

Causal predictive inference and target trial emulation

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