OxCSML seminar this week: Zigzag path connects two Monte Carlo paradigms

[Hai Dang Dau]

Dear all,

Please find below the information for the OxCSML seminar this week. Looking forward to seeing you there.

Best,

Hai-Dang.

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Time and date: Friday 28 Apr 2023, 14:00 to 15:30

Location: Large Lecture Theatre, Department of Statistics, University of Oxford

Speaker: Aki Nishimura, Assistant Professor of Biostatistics, Johns Hopkins University

Title:
Zigzag path connects two Monte Carlo paradigms: Hamiltonian counterparts to piecewise deterministic Markov processes

Zoom registration link: https://www.eventbrite.fr/e/oxcsml-seminar-28-apr-2023-tickets-622410365157

Abstract:
Zigzag and other piecewise deterministic Markov process samplers have attracted significant interest for their non-reversibility and other appealing properties for Bayesian computation. Hamiltonian Monte Carlo is another state-of-the-art sampler, exploiting fictitious momentum to guide Markov chains through complex target distributions.
In this talk, we first establish a remarkable connection between the zigzag sampler and a variant of Hamiltonian Monte Carlo based on Laplace-distributed momentum. The position-velocity component of the corresponding Hamiltonian dynamics travels along a zigzag path paralleling the Markovian zigzag process; however, the dynamics is non-Markovian as the momentum component encodes non-immediate pasts. In the limit of increasingly frequent momentum refreshments in which we preserve its direction but re-sample magnitude, we prove that Hamiltonian zigzag converges strongly to its Markovian counterpart. This theoretical insight in particular explains the two zigzags' relative performance on target distributions with highly correlated parameters, which we demonstrate on a 11,235-dimensional truncated Gaussian target arising from Bayesian phylogenetic multivariate probit model applied to an HIV virus dataset.
We then proceed to construct a Hamiltonian counterpart to the bouncy particle sampler (BPS), further strengthening the connection between the two paradigms. We achieve this by turning BPS's Poisson schedule for velocity switch events into a deterministic one dictated by an auxiliary "inertia" parameter. The resulting Hamiltonian BPS constitutes an effecient sampler on log-concave targets and straightforwadly accommodates parameter contraints. We demonstrate its competitive performance in the posterior computation under Bayesian sparse logistic regression model applied to a large-scale observational study consisting of 72,489 patients and 22,175 clinical covariates.