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A Study on the Properties and Applications of Advanced MCMC Methods for Diffusion Models

Pazos, EA; (2014) A Study on the Properties and Applications of Advanced MCMC Methods for Diffusion Models. Doctoral thesis , UCL (University College London). Green open access

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Abstract

The aim of this thesis is to find ways to make advanced Markov Chain Monte Carlo (MCMC) algorithms more efficient. Our framework is relevant for target distributions defined as change of measures from Gaussian laws; we use this def- inition because it provides the flexibility to apply our methods to a wider range of problems –including models driven by Stochastic Differential Equations (SDE). The advanced MCMC algorithms presented in this thesis are well-defined on the infinite-dimensional path-space and exhibit superior properties in terms of compu- tational complexity. The consequence of the well-definition of these algorithms is that they have mesh-free mixing properties and their convergence time does not de- teriorate when the dimension of the path increases. The contributions we make in this thesis are in four areas: First, we present a new proof for the well-posedness of the advanced Hybrid Monte Carlo (HMC) algorithm; this proof allows us to verify the validity of the required assumptions for well-posedness in several practi- cal applications. Second, by comparing analytically and with numerical examples the computational costs of different algorithms, we show that the advanced Ran- dom Walk Metropolis and the Metropolis-adjusted Langevin algorithm (MALA) have similar complexity when applied to ‘long’ diffusion paths, whereas the HMC algorithm is more efficient than both. Third, we demonstrate that the Golightly- Wikinson transformation can be applied to a wider range of applications – than the typically used Lamperti– when using HMC algorithms to sample from complex target distributions such as SDEs with general diffusion coefficients. Four, we im- plemented a novel joint update scheme to sample from a path observed with error, where the path itself was driven by a fractional Brownian motion (fBm) instead of a Wiener process. Here HMC’s scaling properties proved desirable, since, the non-Markovian properties of fBm made techniques like blocking overly expensive. We achieved this by a well-planned use of the Davies-Harte algorithm to provide the mapping between fBm and uncorrelated white noise that we used to decouple the a-priori involved model parameters from the high-dimensional latent variables. Finally, we showed numerically that our proposed algorithm works efficiently and provided ample comparisons to corroborate it.

Type: Thesis (Doctoral)
Title: A Study on the Properties and Applications of Advanced MCMC Methods for Diffusion Models
Open access status: An open access version is available from UCL Discovery
Language: English
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1457514
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