Computational Complexity Comparison via Scaling Limits for Emerging MCMC Samplers

1 What is a PDMP?

An illustration of a Piecewise Deterministic Markov Process.
Output from the PDMPFlux.jl package.

A Blog Entry on Bayesian Computation by an Applied Mathematician

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1.1 Evolution of Monte Carlo

Markov Chain
(1953–)

Diffusion
(1978–)

PDMP
(2008–)

1.2 Piecewise Deterministic Markov Process

Takeaway: PDMP ≒ ODE + Jump

Langevin Diffusion

Randomized Hamiltonian Monte Carlo

1.3 Which One is Faster ?

Zig-Zag Sampler
(Bierkens et al., 2019)

Bouncy Particle Sampler
(Bouchard-Côté et al., 2018)

Forward Event-Chain Monte Carlo
(Michel et al., 2020)

1.4 ESS Comparison

1.5 ESS: Theory vs. Numerical Results

1.6 New Asymptotic Variance Estimator

References

Bierkens, J., Fearnhead, P., and Roberts, G. (2019). The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data. The Annals of Statistics, 47(3), 1288–1320.

Bouchard-Côté, A., Vollmer, S. J., and Doucet, A. (2018). The Bouncy Particle Sampler: A Nonreversible Rejection-Free Markov Chain Monte Carlo Method. Journal of the American Statistical Association, 113(522), 855–867.

Michel, M., Durmus, A., and Sénécal, S. (2020). Forward event-chain monte carlo: Fast sampling by randomness control in irreversible markov chains. Journal of Computational and Graphical Statistics, 29(4), 689–702.