Symplectic geometry and Hamiltonian Monte Carlo method

Abstract

Hamiltonian Monte Carlo (HMC) method is an application of a non- Euclidean geometry to an inverse problem. HMC is a probabilistic sampling method with the basis of Hamiltonian dynamics. One of the main advantages of HMC algorithm is to draw independent samples from the model space with a higher acceptance rate than other Markov Chain Monte Carlo (MCMC) methods. In order to understand how higher acceptance rate is achieved, I have studied HMC in the light of symplectic geometry. Hamiltonian dynamics is defined on the phase space (cotangent bundle), which has a natural symplectic structure, i.e. a differential two-form which is non-degenerate and closed. Hamiltonian function is defined on the phase space, which corresponds to the sum of misfit and the square of the generalized momentum. By using the non-degeneracy property of symplectic form, a vector field can be found in which Hamiltonian function is invariant along the integral curves of the vector field. The invariance of the Hamil tonian function results in high acceptance rate, where we apply accept-reject test to satisfy detailed-balance property. In this thesis, we define some basic concepts and theorems in symplectic geometry, then describe the relation between symplectic geometry and HMC, namely Hamiltonian dynamics. Lastly, we show an implementation for HMC algorithm to a 2D-tomography problem and analyze the tune parameters for application of HMC.