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The Ohio State University

1. Olsen, Andrew Nolan. When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods.

Degree: PhD, Statistics, 2015, The Ohio State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406

Markov chains are an incredibly powerful tool for
statisticians and other practitioners. They allow for random draws,
though autocorrelated, to be obtained from a vast array of target
distributions, even when the distribution is known only up to a
constant. These draws may then be used to answer key questions of
interest. Markov chains are used in many settings and are the
predominant method for performing inference for Bayesian methods.
The utility of Markov chains lies largely in the simplicity with
which they are implemented. The most basic algorithms are easily
understood and are not challenging to program. The trade-off with
ease of implementation, however, is that issues with Markov chains,
particularly with respect to convergence, can occasionally be left
undiagnosed. For example, a Markov chain may not have been run long
enough to accurately capture the features of the distribution of
interest, or perhaps the error of the resulting estimates is
grossly underrepresented, if it is considered at all.The study of
Markov chain convergence can be summarized by two main
questions:Question 1: Was the simulation run long enough? Question
2: How accurate are the resulting estimates?While simple and clear,
these questions are often left unanswered when Markov chain Monte
Carlo methods are implemented. This is largely due to the fact that
these answers require theoretical analysis of the convergence of
the Markov chain, which can be challenging. This dissertation
discusses the theory of Markov chains and their convergence,
including how to rigorously answer Question 1 and Question 2. A
variety of methods are available, and several are illustrated with
examples.One approach answers Question 1 by obtaining draws that
approximate the target distribution closely. Markov chains may then
be started from these draws, resulting in immediate closeness to
the target distribution. Several algorithms for accomplishing this
are introduced and developed. Results are provided which quantify
the quality of the approximations. A comparison of the efficiency
of the algorithms is also provided. Another approach is the formal
establishment of convergence rates. Once these are established, one
method to answer Question 1 is to compute the number of iterations
required so that the ultimate distribution obtained is close to the
target distribution. This approach is also illustrated with
examples.A final approach is to compute standard errors of the
resulting estimates, which directly answers Question 2. Question 1,
however, is also answered because when estimates are accurate
enough, the chain has been run for a sufficient duration. This is
similarly illustrated with examples.Bayesian scale-usage models are
used to analyze surveys where individual respondents differ in
their use of a rating scale. The convergence rate theory for these
models, which guarantees answers to Question 1 and Question 2, is
fully established. The methods are then extended to a setting where
demographics can govern the way in which respondents differ in
their answer…
*Advisors/Committee Members: Herbei, Radu (Advisor).*

Subjects/Keywords: Statistics; Markov chain Monte Carlo convergence; Markov chain Monte Carlo standard errors; geometric ergodicity; scale-usage heterogeneity

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APA (6^{th} Edition):

Olsen, A. N. (2015). When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406

Chicago Manual of Style (16^{th} Edition):

Olsen, Andrew Nolan. “When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods.” 2015. Doctoral Dissertation, The Ohio State University. Accessed January 19, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406.

MLA Handbook (7^{th} Edition):

Olsen, Andrew Nolan. “When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods.” 2015. Web. 19 Jan 2021.

Vancouver:

Olsen AN. When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods. [Internet] [Doctoral dissertation]. The Ohio State University; 2015. [cited 2021 Jan 19]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406.

Council of Science Editors:

Olsen AN. When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods. [Doctoral Dissertation]. The Ohio State University; 2015. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406