Some Convergence Results for Metropolis-Hastings Algorithms

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Some Convergence Results for Metropolis-Hastings Algorithms

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Brown, Austin

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2022-08

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Abstract

This thesis is concerned with the computational effort required by a Metropolis-Hastings algorithm to converge to the target distribution in total variation and Wasserstein distances. First, under mild assumptions, we show the sharp convergence rate in total variation is also sharp in weaker Wasserstein distances for the Metropolis-Hastings independence sampler. We derive exact convergence expressions for general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together. Next, practitioners are often left tuning Metropolis-Hastings algorithms by trial and error or using optimal scaling guidelines to avoid poor empirical performance. We develop general lower bounds on the convergence rates of geometrically ergodic Metropolis-Hastings algorithms to study their computational complexity. If the target density concentrates with a parameter n (e.g. Bayesian posterior concentration, Laplace approximations), we show the convergence rate can tend to 1 exponentially fast if the tuning parameters do not depend carefully on the dimension and the parameter n.

Keywords

Convergence analysis

Markov chain Monte Carlo

Metropolis-Hastings

Description

University of Minnesota Ph.D. dissertation. 2022. Major: Statistics. Advisor: Galin Jones. 1 computer file (PDF); 115 pages.

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Brown, Austin. (2022). Some Convergence Results for Metropolis-Hastings Algorithms. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/243073.

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