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Markov Chain Monte Carlo in Practice.
TLDR
Methods for assessing the reliability of the simulation effort are reviewed, with an emphasis on those most useful in practically relevant settings.
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Geometric convergence bounds for Markov chains in Wasserstein distance based on generalized drift and contraction conditions
  • Qian Qin, J. Hobert
  • Mathematics
    Annales de l'Institut Henri Poincaré, Probabilit…
  • 8 February 2019
Let $\{X_n\}_{n=0}^\infty$ denote an ergodic Markov chain on a general state space that has stationary distribution $\pi$. This article concerns upper bounds on the $L_1$-Wasserstein distance between…
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Estimating the spectral gap of a trace-class Markov operator
The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating)…
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Wasserstein-based methods for convergence complexity analysis of MCMC with applications
TLDR
Results presented herein suggest that d&m may be less useful in the emerging area of convergence complexity analysis, which is the study of how the convergence behavior of Monte Carlo Markov chains scale with sample size, sample size and number of covariates.
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Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression
  • Qian Qin, J. Hobert
  • Computer Science, Mathematics
    The Annals of Statistics
  • 24 December 2017
TLDR
The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions.
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On the limitations of single-step drift and minorization in Markov chain convergence analysis
TLDR
It is shown that any convergence rate bound based on a set of d-m conditions cannot do better than a certain unknown optimal bound, and the results strongly suggest that convergence rate bounds based on single-step d\&m conditions are quite inadequate in high-dimensional settings.
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Asymptotically Stable Drift and Minorization for Markov Chains with Application to Albert and Chib's Algorithm
TLDR
The key ideas include developing an appropriately "centered" drift function, and suppressing high-dimensionality in the construction of minorization conditions, which are employed in a thorough convergence complexity analysis of Albert and Chib's (1993) data augmentation algorithm for the Bayesian probit model.
Convergence Analysis of MCMC Algorithms for Bayesian Multivariate Linear Regression with Non‐Gaussian Errors
When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals.…
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Trace-class Monte Carlo Markov chains for Bayesian multivariate linear regression with non-Gaussian errors
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Wasserstein-based methods for convergence complexity analysis of MCMC with application to Albert and Chib's algorithm
TLDR
Results presented herein suggest that d&m may be less useful in the emerging area of convergence complexity analysis, which is the study of how Monte Carlo Markov chain convergence behavior scales with sample size, sample size and/or number of covariates.
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