The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. We show that positive curvature can be used to prove theoretical concentration results for HMC Markov chains.
Christof Seiler (Stanford University)
Simon Rubinstein-Salzedo (Stanford University)
Susan Holmes (Stanford University)
Brought up in the French School of Data Analysis (Analyse des Données) in the 1980's, Professor Holmes specializes in exploring and visualizing complex biological data. She is interested in integrating the information provided by phylogenetic trees, community interaction graphs and metabolic networks with sequencing data and clinical covariates. She uses computational statistics, and Bayesian methods to draw inferences about many complex biological phenomena such as the human microbiome or the interactions between the immune system and cancer. She teaches using R and BioConductor and tries to make everything she does freely available.
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