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

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.