Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm
Leif T. Johnson, Charles J. Geyer
The Annals of Statistics · 2012 · Volume 40 · Issue 6
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.
Keywords
60J05, 60J22, 65C05, change of variable, conjugate prior, drift condition, exponential family, Markov chain isomorphism, Markov chain Monte Carlo, Metropolis–Hastings–Green algorithm
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