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On Hamiltonian Monte Carlo for Gaussian Random Variables with Random Hamiltonians

Yingdong Lu, Tomasz Nowicki

Abstract

We study a family of (multivariate-)Gaussian Hamiltonian Monte Carlo (GHMC) operators and prove that the family of Gaussian distributions and their mixtures are invariant under such operators. Furthermore, each such operator is a contraction on the space of parameters and an explicit formulae are derived. These results then enable us to analyze the dynamics and convergences of independent and identically distributed random sequences of such operators.

On Hamiltonian Monte Carlo for Gaussian Random Variables with Random Hamiltonians

Abstract

We study a family of (multivariate-)Gaussian Hamiltonian Monte Carlo (GHMC) operators and prove that the family of Gaussian distributions and their mixtures are invariant under such operators. Furthermore, each such operator is a contraction on the space of parameters and an explicit formulae are derived. These results then enable us to analyze the dynamics and convergences of independent and identically distributed random sequences of such operators.
Paper Structure (10 sections, 14 theorems, 62 equations)

This paper contains 10 sections, 14 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian Monte Carlo for Gaussian Target and Auxiliary
  3. The general case in $\mathbb R^d$
  4. Hamiltonian Monte Carlo, Gaussian edition
  5. Invariance and contraction, one step
  6. Metric induced by the HMC for univariate Gaussians
  7. GHMC with Random Potentials
  8. Convex limit set when targets may vary
  9. The General Multivariate Case
  10. Power series with i.i.d. coefficients, convergence in the univariate case

Key Result

Proposition 1

For GHMC with Hamiltonian function ${\cal H}(Q,P)=\frac{1}{2}\left((Q-\mu_\mathfrak{f})^\top F (Q-\mu_\mathfrak{f})+(P-\mu_\mathfrak{g})^\top G (P-\mu_\mathfrak{g})\right)+{\rm const\,}$, the Hamiltonian motion with time parameter $t$ is given by: where we used the notation eqn:notation.

Theorems & Definitions (29)

  • Remark
  • Remark 1: Properties of covariances
  • Proposition 1: Gaussian motion
  • proof
  • Corollary 1: GHMC acting on ${\cal G}(d)$
  • Theorem 1
  • Lemma 1: Representation of a sum of quadratic terms
  • Remark 2
  • proof
  • Corollary 2: New Gaussian distribution
  • ...and 19 more