Convergence rates for a hierarchical Gibbs sampler

DOI10.3150/15-BEJ758zbMATH Open1379.60083arXiv1402.4733MaRDI QIDQ502894

Neal Madras, Oliver Jovanovski

Publication date: 11 January 2017

Published in: Bernoulli (Search for Journal in Brave)

Abstract: We establish some results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate.

Full work available at URL: https://arxiv.org/abs/1402.4733

zbMATH Keywords

Markov chainconvergence rategamma distributioncouplingstochastic monotonicityhierarchical Gibbs sampler

Mathematics Subject Classification ID

Computational methods in Markov chains (60J22) Bayesian inference (62F15)

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