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MCMC Algorithms for Posteriors on Matrix Spaces

Beskos, A; Kamatani, K; (2022) MCMC Algorithms for Posteriors on Matrix Spaces. Journal of Computational and Graphical Statistics , 31 (3) pp. 721-738. 10.1080/10618600.2022.2058953. Green open access

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Abstract

We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank–Nicolson (pCN), a contribution of this paper in the development of a novel algorithm, termed the ‘Mixed’ pCN (MpCN). RWM and pCN are shown not to be geometrically ergodic for an important class of matrix distributions with heavy tails. In contrast, MpCN is robust across targets with different tail behaviour and has very good empirical performance within the class of heavy-tailed distributions. Geometric ergodicity for MpCN is not fully proven in this work, as some remaining drift conditions are quite challenging to obtain owing to the complexity of the state space. We do, however, make a lot of progress towards a proof, and show in detail the last steps left for future work. We illustrate the computational performance of the various algorithms through numerical applications, including calibration on real data of a challenging model arising in financial statistics.

Type: Article
Title: MCMC Algorithms for Posteriors on Matrix Spaces
Open access status: An open access version is available from UCL Discovery
DOI: 10.1080/10618600.2022.2058953
Publisher version: https://doi.org/10.1080/10618600.2022.2058953
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.
Keywords: Preconditioned Crank–Nicolson, Drift Condition, Matrix-Valued Stochastic Differential Equation
UCL classification: UCL
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/10137877
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