Perfecting MCMC Sampling: Recipes and Reservations

This review paper is intended for the Handbook of Markov chain Monte Carlo's second edition. The authors will be grateful for any suggestions that could perfect it

Computing Bayes: From Then 'Til Now'

This paper takes the reader on a journey through the history of Bayesian computation, from the 18th century to the present day. Beginning with the one-dimensional integral first confronted by Bayes in 1763, we highlight the key contributions of: Laplace, Metropolis (and, importantly, his co-authors!), Hammersley and Handscomb, and Hastings, all of which set the foundations for the computational revolution in the late 20th century -- led, primarily, by Markov chain Monte Carlo (MCMC) algorithms. A very short outline of 21st century computational methods -- including pseudo-marginal MCMC, Hamiltonian Monte Carlo, sequential Monte Carlo, and the various `approximate' methods -- completes the paper.Comment: Material that appeared in an earlier paper, `Computing Bayes: Bayesian Computation from 1763 to the 21st Century' (arXiv:2004.06425) has been broken up into two separate papers: this historical overview of, and timeline for, all computational developments is retained; and a secondary paper (arXiv:2112.10342), which provides a more detailed review of 21st centur

ε-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

Consider a fractional Brownian motion (fBM) BH={BH(t):t∈[0,1]} with Hurst index H∈(0,1). We construct a probability space supporting both BH and a fully simulatable process B⌢Hε such that supt∈[0,1]∣∣∣BH(t)−B⌢H∈(t)∣∣∣≤ε with probability one for any user-specified error bound ɛ>0. When H>1/2, we further enhance our error guarantee to the α-Hölder norm for any α∈(1/2,H). This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations Y={Y(t):t∈[0,1]}. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process Y⌢ε such that supt∈[0,1]∣∣Y(t)−Y⌢ε(t)∣∣≤ε with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently