Bayesian consistency for a nonparametric stationary Markov model

Chae, Minwoo; Walker, Stephen G.

doi:10.3150/17-bej1007

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…Müller et al (1997) use normalized weights that employ a finite MoE framework to accommodate nonlinearity. Posterior consistency for BNP transition density estimation has been explored by Tang and Ghosal (2007a), Tang and Ghosal (2007b), and Chae and Walker (2019). Many of the above methods assume first-order time dependence.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Nonparametric Density Autoregression with Lag Selection

Heiner

Kottas

2022

Bayesian Anal.

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We develop a Bayesian nonparametric autoregressive model applied to flexibly estimate general transition densities exhibiting nonlinear lag dependence. Our approach is related to Bayesian density regression using Dirichlet process mixtures, with the Markovian likelihood defined through the conditional distribution obtained from the mixture. This results in a Bayesian nonparametric extension of a mixtures-of-experts model formulation. We address computational challenges to posterior sampling that arise from the Markovian structure in the likelihood. The base model is illustrated with synthetic data from a classical model for population dynamics, as well as a series of waiting times between eruptions of Old Faithful Geyser. We study inferences available through the base model before extending the methodology to include automatic relevance detection among a pre-specified set of lags. Inference for global and local lag selection is explored with additional simulation studies, and the methods are illustrated through analysis of an annual time series of pink salmon abundance in a stream in Alaska. We further explore and compare transition density estimation performance for alternative configurations of the proposed model. Supplementary materials are available online.

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Section: Introductionmentioning

confidence: 99%

Bayesian Nonparametric Density Autoregression with Lag Selection

Heiner

Kottas

2022

Bayesian Anal.

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“…Recently, [3] proposed the use of the Wasserstein distance in the implementation of Approximate Bayesian Computation (ABC) to approximate the posterior distribution. In nonparametric Bayesian inference, [36,12] used Wasserstein metrics to study asymptotic properties of posterior distributions, but W p was considered as a distance between mixing distributions rather than a distance between mixture densities themselves. As a result, the Wasserstein metrics in these papers yielded a stronger topology than the total variation distance on the space of density functions.…”

Section: Introductionmentioning

confidence: 99%

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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“…Recently, Bernton et al (2019) proposed the use of the Wasserstein distance in the implementation of Approximate Bayesian Computation (ABC) to approximate the posterior distribution. In nonparametric Bayesian inference, Chae and Walker (2019a), Nguyen (2013) used Wasserstein metrics to study asymptotic properties of posterior distributions, but W p was considered as a distance between mixing distributions rather than a distance between mixture densities themselves. As a result, the Wasserstein metrics in these papers yielded a stronger topology than the total variation distance on the space of density functions.…”

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.

show abstract

Bayesian consistency for a nonparametric stationary Markov model

Cited by 6 publications

References 31 publications

Bayesian Nonparametric Density Autoregression with Lag Selection

Bayesian Nonparametric Density Autoregression with Lag Selection

Posterior asymptotics in Wasserstein metrics on the real line

Posterior asymptotics in Wasserstein metrics on the real line

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