The Annals of Statistics

Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm

Leif T. Johnson and Charles J. Geyer

Full-text: Open access

Abstract

A random-walk Metropolis sampler is geometrically ergodic if its equilibrium density is super-exponentially light and satisfies a curvature condition [Stochastic Process. Appl. 85 (2000) 341–361]. Many applications, including Bayesian analysis with conjugate priors of logistic and Poisson regression and of log-linear models for categorical data result in posterior distributions that are not super-exponentially light. We show how to apply the change-of-variable formula for diffeomorphisms to obtain new densities that do satisfy the conditions for geometric ergodicity. Sampling the new variable and mapping the results back to the old gives a geometrically ergodic sampler for the original variable. This method of obtaining geometric ergodicity has very wide applicability.

Article information

Source
Ann. Statist., Volume 40, Number 6 (2012), 3050-3076.

Dates
First available in Project Euclid: 22 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1361542074

Digital Object Identifier
doi:10.1214/12-AOS1048

Mathematical Reviews number (MathSciNet)
MR3097969

Zentralblatt MATH identifier
1302.46033

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 65C05: Monte Carlo methods
Secondary: 60J22: Computational methods in Markov chains [See also 65C40]

Keywords
Markov chain Monte Carlo change of variable exponential family conjugate prior Markov chain isomorphism drift condition Metropolis–Hastings–Green algorithm

Citation

Johnson, Leif T.; Geyer, Charles J. Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm. Ann. Statist. 40 (2012), no. 6, 3050--3076. doi:10.1214/12-AOS1048. https://projecteuclid.org/euclid.aos/1361542074


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