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Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation

Holbrook, Andrew and Lan, Shiwei and Vandenberg-Rodes, Alexander and Shahbaba, Babak (2017) Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation. Journal of Statistical Computation and Simulation, 88 (5). pp. 982-1002. ISSN 0094-9655. https://resolver.caltech.edu/CaltechAUTHORS:20180207-141759462

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Abstract

We present geodesic Lagrangian Monte Carlo, an extension of Hamiltonian Monte Carlo for sampling from posterior distributions defined on general Riemannian manifolds. We apply this new algorithm to Bayesian inference on symmetric or Hermitian positive definite (PD) matrices. To do so, we exploit the Riemannian structure induced by Cartan's canonical metric. The geodesics that correspond to this metric are available in closed-form and – within the context of Lagrangian Monte Carlo – provide a principled way to travel around the space of PD matrices. Our method improves Bayesian inference on such matrices by allowing for a broad range of priors, so we are not limited to conjugate priors only. In the context of spectral density estimation, we use the (non-conjugate) complex reference prior as an example modelling option made available by the algorithm. Results based on simulated and real-world multivariate time series are presented in this context, and future directions are outlined.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1080/00949655.2017.1416470DOIArticle
http://www.tandfonline.com/doi/full/10.1080/00949655.2017.1416470PublisherArticle
https://arxiv.org/abs/1612.08224arXivDiscussion Paper
ORCID:
AuthorORCID
Holbrook, Andrew0000-0002-3558-200X
Lan, Shiwei0000-0002-9167-3715
Additional Information:© 2017 Taylor & Francis. Received 25 Jul 2017, Accepted 09 Dec 2017, Published online: 27 Dec 2017. AH is supported by NIH grant [T32 AG000096]. SL is supported by the Defense Advanced Research Projects Agency (DARPA) funded program Enabling Quantification of Uncertainty in Physical Systems (EQUiPS), contract W911NF-15-2-0121. AV and BS are supported by National Institutes of Health [grant R01-AI107034] and National Science Foundation [grant DMS-1622490].
Funders:
Funding AgencyGrant Number
NIH Predoctoral FellowshipT32 AG000096
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Army Research Office (ARO)W911NF-15-2-0121
NIHR01-AI107034
NSFDMS-1622490
Subject Keywords:HMC, Riemannian geometry, spectral analysis
Issue or Number:5
Record Number:CaltechAUTHORS:20180207-141759462
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180207-141759462
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:84728
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 Feb 2018 00:12
Last Modified:03 Oct 2019 19:21

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