A novel approach to Bayesian consistency

Chae, Minwoo; Walker, Stephen G.

doi:10.1214/17-ejs1369

Cited by 8 publications

(12 citation statements)

References 28 publications

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“…as used by Chae and Walker [2017]. We can now determine that sup is the Kolmogorov distance where F and F 0 are cumulative distribution functions of f and f 0 .…”

Section: Discussionmentioning

confidence: 99%

Bayesian Consistency with the Supremum Metric

Ho¹,

Walker²

2022

Preprint

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We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback-Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure L 1 consistency.

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“…as used by Chae and Walker [2017]. We can now determine that sup is the Kolmogorov distance where F and F 0 are cumulative distribution functions of f and f 0 .…”

Section: Discussionmentioning

confidence: 99%

Bayesian Consistency with the Supremum Metric

Ho¹,

Walker²

2022

Preprint

Self Cite

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“…This is a certain reverse inequality because the total variation generates a stronger topology than W p in the space of all probability measures on a bounded metric space. This kind of reverse inequality and related theory for posterior consistency are the main motivation of the present paper, which was firstly considered in [11].…”

Section: Fig 2 Wasserstein Distances Between the True Distribution-uniform (Left) Normal (Middle) And Laplace (Right)-and The Posterior Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…On the other hand, it is surprising that careful analysis of the convergence rates with respect to a weak metric such as the Lévy-Prokhorov and Kolmogorov has not been studied in literature, considering that the KL support condition is sufficient for the consistency in those metrics. [11] studied the convergence rate of the posterior distribution with respect to the Lévy-Prokhorov metric, but their rate n −1/4 have a lot of room for improvement. Furthermore, they used the Lévy-Prokhorov rate as a tool for proving the consistency in total variation, and did not focus on the convergence rate itself.…”

Section: Introductionmentioning

confidence: 99%

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Posterior asymptotics in Wasserstein metrics on the real line

Chae

Blasi

Walker

2021

Electron. J. Statist.

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In this paper, we use the class of Wasserstein metrics to study asymptotic properties of posterior distributions. Our first goal is to provide sufficient conditions for posterior consistency. In addition to the well-known Schwartz's Kullback-Leibler condition on the prior, the true distribution and most probability measures in the support of the prior are required to possess moments up to an order which is determined by the order of the Wasserstein metric. We further investigate convergence rates of the posterior distributions for which we need stronger moment conditions. The required tail conditions are sharp in the sense that the posterior distribution may be inconsistent or contract slowly to the true distribution without these conditions. Our study involves techniques that build on recent advances on Wasserstein convergence of empirical measures. We apply the results to some examples including a Dirichlet process mixture prior and conduct a simulation study for further illustration.

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“…In literature, studies on posterior asymptotics have focused on strong metrics such as the total variation, Hellinger and uniform metrics. For those purposes, some non-trivial conditions such as the bounded entropy or prior summability are assumed in addition to the KL conditions, see Barron et al (1999), Chae and Walker (2017), Ghosal et al (1999), Walker (2004) for example. On the other hand, it is surprising that careful analysis of the convergence rates with respect to a weak metric such as the Lévy-Prokhorov and Kolmogorov has not been studied in literature, considering that the KL support condition is sufficient for the consistency in those metrics.…”

Section: Introductionmentioning

confidence: 99%

Posterior asymptotics in Wasserstein metrics on the real line

Chae

Blasi

Walker

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we use the class of Wasserstein metrics to study asymptotic properties of posterior distributions. Our first goal is to provide sufficient conditions for posterior consistency. In addition to the well-known Schwartz's Kullback-Leibler condition on the prior, the true distribution and most probability measures in the support of the prior are required to possess moments up to an order which is determined by the order of the Wasserstein metric. We further investigate convergence rates of the posterior distributions for which we need stronger moment conditions. The required tail conditions are sharp in the sense that the posterior distribution may be inconsistent or contract slowly to the true distribution without these conditions. Our study involves techniques that build on recent advances on Wasserstein convergence of empirical measures. We apply the results to density estimation with a Dirichlet process mixture prior and conduct a simulation study for further illustration.

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A novel approach to Bayesian consistency

Cited by 8 publications

References 28 publications

Bayesian Consistency with the Supremum Metric

Bayesian Consistency with the Supremum Metric

Posterior asymptotics in Wasserstein metrics on the real line

Posterior asymptotics in Wasserstein metrics on the real line

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