Book chapter
Sacred and Profane: From the Involutive Theory of MCMC to Helpful Hamiltonian Hacks
Handbook of Markov Chain Monte Carlo, pp 418-442
2026
Abstract
Geyer and Neal's landmark chapters in the First Edition of this Handbook provide distinct yet seemingly related views on MCMC algorithms that leverage deterministic dynamics. We review a recently established theory of involutive MCMC and show that this theory provides a natural measure theoretic context that ties together the Metropolis-Hastings-Green and Hamiltonian Monte Carlo (HMC) algorithms respectively discussed by Geyer and Neal. We show that the involutive theory also provides rigorous bases for deterministic MCMC over general (infinite-dimensional) state spaces and surrogate-trajectory HMC algorithms that use approximate dynamics but leave targets invariant. Far from being an end-all be-all theory of MCMC, the involutive theory also provides opportunities for rigorous algorithmic development, a fact we suggest here using a recently established theory of multiproposal MCMC. We illustrate these methods with examples from Bayesian phylogenetics, dimension reduction and inverse problems, emphasizing implementational considerations and scope for future research.
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Details
- Title
- Sacred and Profane: From the Involutive Theory of MCMC to Helpful Hamiltonian Hacks
- Creators
- Nathan E. Glatt-Holtz - Indiana University BloomingtonAndrew J. Holbrook - University of California, Los AngelesJustin A. Krometis - Virginia TechCecilia F. Mondaini - Drexel UniversityAmi Sheth - University of California, Los Angeles
- Contributors
- Radu V. Craiu (Editor)Dootika Vats (Editor)Galin L. Jones (Editor)Steve Brooks (Editor)Andrew Gelman (Editor)Xiao-Li Meng (Editor)
- Publication Details
- Handbook of Markov Chain Monte Carlo, pp 418-442
- Publisher
- CRC Press
- Edition
- 2
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022164638004721