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Title: Poisson equation and weak approximation for Metropolis-Hastings chains
Author: Vogrinc, Jure
ISNI:       0000 0004 6496 9486
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2018
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The work presented investigates speeding up MCMC methods by introducing control variates based on approximate solutions of the Poisson equation. In the setting of Metropolis-Hastings chains in Rd two scalable approaches of approximately solving the Poisson equation are discussed. In both cases an underlying weakly convergent sequence of related Markov chains, enumerated by a scaling parameter, is identi ed and results, asymptotic in the scaling parameter, are given for the achieved improvement. In the rst approach control variates are constructed according to a sequence of ner and ner partitions of the state-space of the Metropolis-Hastings chain, with the mesh of the partition serving as the scaling parameter. In this context it is shown, that as the mesh reduces arbitrarily, so does the asymptotic variance in the Central limit theorem associated with the control variate given by the partition. The second approach assumes a target density of a product type and scales the dimension of the state-space and the variance of the proposal simultaneously. The resulting weakly convergent sequence converges to a Langevin di usion, which is then used to construct control variates for the Metropolis-Hastings chains in the sequence. The bounds obtained in this context suggest the improvement achieved by this approach grows almost linearly in dimension.
Supervisor: Mijatović, Alexandar Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral