Advanced MCMC methods for sampling on diffusion pathspace

Abstract

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte-Carlo methods. We study here an advanced version of familiar Markov-chain Monte-Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate its computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinitedimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

“…In addition, the field of pseudo-Riemannian geometry deals with forms of G(x), which need not be positive-definite [39]; so again, understanding could be gained from here. Some recent work in high-dimensional inference has centred on defining MCMC methods for which efficiency scales O(1) with respect to the dimension, n, of π(·) [19,59]. In the case where X takes values in some infinite-dimensional function space, this can be done provided a Gaussian prior measure is defined for X.…”

“…The key challenge for MCMC is to define transition kernels for which proposed moves are inside the support for π(·). A straight-forward approach is to define proposals for which the prior is invariant, since the likelihood contribution to the posterior typically will not alter its support from that of the prior [19]. However, the posterior may still look very different from the prior, as noted in [61], so this proposal mechanism, though O(1), can still result in slow exploration.…”

“…Increasing the former will reduce the autocorrelation in the chain if the proposal is accepted, but if it is rejected, the chain will not move at all, so autocorrelation will be high. Random walk proposals are sometimes referred to as blind (e.g., [19]), as no information about π(·) is used when generating proposals, so typically, very large moves will result in a very low chance of acceptance, while small moves will be accepted, but result in very high autocorrelation for the chain. Several authors have also shown that for certain classes of π(·), the tuning parameter, λ, should be chosen, such that λ 2 ∝ n −1 , so that α 0 as n → ∞ [20].…”

Bayesian Inference for Diffusion-Driven Mixed-Effects Models

“…Moreover, the optimal value (in terms of maximising acceptance rate) may vary between observation intervals. Beskos et al (2013) use Hybrid Monte Carlo (HMC) on pathspace to generate SDE sample paths under various observation regimes. For the applications considered, the authors found reasonable gains in overall efficiency (as measured by minimum effective sample size per CPU time) over an independence sampler with a Brownian bridge proposal.…”

Bayesian inference for partially observed stochastic differential equations driven by fractional Brownian motion

“…Standard hybrid Monte Carlo algorithm We use the hybrid Monte Carlo algorithm to explore the posterior of Z, θ in (6). The standard method was introduced in Duane et al (1987), but we employ an advanced version, tailored to the structure of the distributions of interest and closely related to algorithms developed in Beskos et al (2011Beskos et al ( , 2013a for effective sampling of change of measures from Gaussian laws in infinite dimensions. We first briefly describe the standard algorithm.…”