On the Application of the Diaconis-Holmes-Neal Sampler to Unimodal Underlying Probabilities

Author

Christopher J. Lange, University at Albany, State University of New YorkFollow

Date of Award

Spring 2025

Language

English

Embargo Period

4-10-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Program

Mathematics

First Advisor

Martin Hildebrand

Committee Members

Karin Reinhold, Josh Isralowitz, Felix Ye

Abstract

Upon the introduction of the Metropolis et. al (1953) [6] algorithm, the question of how many steps in the Markov chain were needed to achieve convergence to stationarity became apparent. Eventually it was found that the convergence was rather slow, i.e. for a process on n states the number of steps needed to achieve convergence to stationarity was found to be on the order of n². [8]

The obvious problem with Metropolis et. al (1953) is that the Markov chain is reversible. [6] In other words, for any state j we can move from j to j + 1 and back to j in two steps. To correct for this, Diaconis, Holmes, and Neal (2000) improved Metropolis et. al (1953) by introducing a non-reversible Markov chain [8]. The Diaconis-Holmes-Neal sampler, as it is widely known, is a Markov chain on two copies of n states, a +1 copy and a −1 copy. The fact that the sampler includes two copies of n states also solves the not-as-serious problem of Metropolis et. al only acting on one copy of n states.

Applications of the Diaconis-Holmes-Neal sampler include Markov chain sampling and situations in statistical physics, among others [8]. However, an answer to the question of how many steps are needed to achieve convergence to stationarity was required. In a 2002 preprint, Hildebrand showed that if the underlying probabilities are log-concave then the sampler achieves convergence to stationarity in at least a constant multiple of n steps [3].

Nonetheless, the question of whether a similar convergence exists when the underlying probabilities are instead unimodal was posed in the preprint. In this dissertation, we answer the question in the two most general cases, namely the general symmetric unimodal case (also known as the general symmetric case) and the general unimodal case.

License

This work is licensed under the University at Albany Standard Author Agreement.

Recommended Citation

Lange, Christopher J., "On the Application of the Diaconis-Holmes-Neal Sampler to Unimodal Underlying Probabilities" (2025). Electronic Theses & Dissertations (2024 - present). 127.
https://scholarsarchive.library.albany.edu/etd/127