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Published December 27, 2017 | Submitted
Journal Article Open

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

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.

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].

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