2020/04/29 by Bryant Davis, Davis, Bryant, James P. Hobert +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2004.14330
openalex publication_date 2020/04/29 · openalex created_date 2022/07/19 · openalex updated_date 2026/07/28
The emergence of big data has led to so-called convergence complexity\nanalysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms\nbehave as the sample size, n, and/or the number of parameters, p, in the\nunderlying data set increase. This type of analysis is often quite challenging,\nin part because existing results for fixed n and p are simply not sharp\nenough to yield good asymptotic results. One of the first convergence\ncomplexity results for an MCMC algorithm on a continuous state space is due to\nYang and Rosenthal (2019), who established a mixing time result for a Gibbs\nsampler (for a simple Bayesian random effects model) that was introduced and\nstudied by Rosenthal (1996). The asymptotic behavior of the spectral gap of\nthis Gibbs sampler is, however, still unknown. We use a recently developed\nsimulation technique (Qin et. al., 2019) to provide substantial numerical\nevidence that the gap is bounded away from 0 as n \→ \∞. We also\nestablish a pair of rigorous convergence complexity results for two different\nGibbs samplers associated with a generalization of the random effects model\nconsidered by Rosenthal (1996). Our results show that, under strong regularity\nconditions, the spectral gaps of these Gibbs samplers converge to 1 as the\nsample size increases.\n