Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 603-625.

Convergence rates for a hierarchical Gibbs sampler

Oliver Jovanovski and Neal Madras

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Abstract

We establish results for the rate of convergence in total variation of a particular Gibbs sampler to its equilibrium distribution. This sampler is for a Bayesian inference model for a gamma random variable, whose only complexity lies in its multiple levels of hierarchy. Our results apply to a wide range of parameter values when the hierarchical depth is 3 or 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 coordinates. We construct auxiliary stochastic processes to show that this ratio converges to 1 at a geometric rate.

Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 603-625.

Dates
Received: October 2014
Revised: July 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001368

Digital Object Identifier
doi:10.3150/15-BEJ758

Mathematical Reviews number (MathSciNet)
MR3556786

Zentralblatt MATH identifier
1379.60083

Keywords
convergence rate coupling gamma distribution hierarchical Gibbs sampler Markov chain stochastic monotonicity

Citation

Jovanovski, Oliver; Madras, Neal. Convergence rates for a hierarchical Gibbs sampler. Bernoulli 23 (2017), no. 1, 603--625. doi:10.3150/15-BEJ758. https://projecteuclid.org/euclid.bj/1475001368


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