Estimation of Bayes Factors in a Class of Hierarchical Random Effects Models Using a Geometrically Ergodic MCMC Algorithm

Abstract

We consider a Bayesian random effects model that is commonly used in meta-analysis, in which the random effects have a t distribution, with degrees of freedom parameter to be estimated. We develop a Markov chain Monte Carlo algorithm for estimating the posterior distribution in this model, and establish geometric convergence of the algorithm. The geometric convergence rate has important theoretical and practical ramifications. Indeed, it implies that, under standard second moment conditions, the ergodic averages used to estimate posterior quantities of interest satisfy central limit theorems. Moreover, it guarantees the consistency of a batch means estimate of the asymptotic variance in the CLT, which in turn allows for the construction of asymptotically valid standard errors. We show how our Markov chain can be used, in conjunction with an importance sampling method, to carry out an empirical Bayes approach for estimating the degrees of freedom parameter. To illustrate our methodology we consider a meta-analysis of studies that link intake of non-steroidal anti-inflammatory drugs to a reduction in colon cancer risk, in which some of the studies are outliers. To model the distribution of the study effects we consider the family of t distributions, as well as a family of mixtures of Dirichlet process priors centered at the t distributions, and show how our methodology can be used to make a choice of model.

Geometric ergodicity of random scan Gibbs samplers for hierarchical one-way random effects models

“…Interestingly, the conditions on the sample size in our Theorem 1 are slightly weaker than those obtained by either Hobert and Geyer (1998) or Jones and Hobert (2004) for the deterministic scan Gibbs sampler. Doss and Hobert (2010) consider the setting of section 2.2 and prove that the deterministic scan Gibbs sampler is geometrically ergodic when a 2 = b 2 = d/2, d ≥ 1 and 2a…”

“…Doss and Hobert (2010) consider a hierarchical model which is a special case of the model presented in the previous section, that is, (5). Let λ = (λ 1 , .…”

“…Also, m 0 ∈ R is known. Doss and Hobert (2010) use this model in the context of meta-analysis where it can be reasonable to assume the γ i are known. The posterior density is f (ξ, λ θ , λ|y), yielding full conditional distributions…”

“…The argument is the same as one given by Doss and Hobert (2010) in their proof of geometric ergodicity of the deterministic scan Gibbs sampler, but is included here for the sake of completeness since their drift function did not have our w 8 , w 9 and w 10 . Let T > 0 be arbitrary and observe that w 5 → ∞ and w 9 → ∞ as λ θ → ∞ while w 1 → ∞ as λ θ → 0; w 6 → ∞ and w 10 → ∞ as λ i → ∞ while w 2 → ∞ as λ i → 0; w 4 → ∞ as |θ i | → ∞; and conditional on θ i ∈ L T we see that w 3 → ∞ and w 8 → ∞ as |µ| → ∞.…”

“…There has been a fair amount of work on establishing geometric ergodicity of two-component deterministic scan Gibbs samplers (see, among others, Hobert and Geyer, 1998;Jones and Hobert, 2004;Marchev and Hobert, 2004;Papaspiliopoulos and Roberts, 2008;Roberts and Polson, 1994;Roberts and Rosenthal, 1999;Roy and Hobert, 2007;Tan and Hobert, 2012), while Doss and Hobert (2010) and Jones and Hobert (2004) considered geometric ergodicity of three-component deterministic scan Gibbs samplers. However, there has been almost no corresponding investigation for random scan Gibbs samplers.…”