View Proposal

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Proposer

Wai Meng Kwok

Title

Investigating the Hamiltonian Monte Carlo algorithm in comparison to the Random Walk Metropolis-Hastings algorithm

Goal

To investigate the Hamiltonian Monte Carlo algorithm and compare it to the Random Walk Metropolis-Hastings algorithm via simulation studies and subsequent applications to real datasets.

Description

Markov Chain Monte Carlo (MCMC) algorithms seek to sample from a (often complex) target probability distribution by constructing an ergodic Markov Chain over the parameter space by proposing new states with suitable acceptance probabilities. A common example is the Random Walk Metropolis-Hastings (RWMH) algorithm, which utilizes symmetric proposals based on the most recent sample of the chain. In low dimensions, the RWMH algorithm performs sufficiently well but its performance quickly deteriorates as the number of dimensions increase, since the region of high probability density becomes relatively “narrower” in high dimensions. The Hamiltonian Monte Carlo (HMC) algorithm attempts to overcome such issues by traversing through the high-dimensional parameter spaces more efficiently. Its proposals are inspired by statistical mechanics and utilizes the gradient information of the target probability distribution to focus on high-probability regions of the parameter space. In this project, the student would first outline the complete HMC algorithm and the various hyperparameters used to tune the algorithm, then examine its sampling capabilities in relation to the RWMH algorithm in several settings – unimodal and bimodal distributions in low-dimensional and high-dimensional settings. Similar methodologies can then be applied on to simulated and real datasets (e.g. inferring regression model coefficients) with appropriate diagnostics.

Resources

1. Betancourt, Michael. "A conceptual introduction to Hamiltonian Monte Carlo." arXiv preprint arXiv:1701.02434 (2017). 2. Gelman, Andrew. Bayesian Data Analysis. Chapman & Hall/Crc, 2004. 3. Yamada, Taisuke, Keitaro Ohno, and Yusaku Ohta. "Comparison between the Hamiltonian Monte Carlo method and the Metropolis–Hastings method for coseismic fault model estimation." Earth, Planets and Space 74.1 (2022): 86. 4. Betancourt, Michael. "The convergence of Markov chain Monte Carlo methods: from the Metropolis method to Hamiltonian Monte Carlo." Annalen der Physik 531.3 (2019): 1700214.

Background

Url

Difficulty Level

Moderate

Ethical Approval

None

Number Of Students

1

Supervisor

Wai Meng Kwok

Keywords

hamiltonian monte carlo, bayesian inference, sampling

Degrees

Bachelor of Science in Statistical Data Science