On the variance reduction of Hamiltonian Monte Carlo via an approximation scheme

Let π be a probability distribution in Rᵈ and f a test function, and consider the problem of variance reduction in estimating E_π(f) . We first construct a sequence of estimators for E_π (f) , say (1/k)∑ᵢ₌₀ᵏ⁻¹ gₙ(Xᵢ) , where the Xᵢ are samples from π generated by the Metropolized Hamiltonian Monte Carlo algorithm and gₙ is the approximate solution of the Poisson equation through the weak approximate scheme recently invented by Mijatović and Vogrinc (2018). Then we prove under some regularity assumptions that the estimation error variance σ_π²(gₙ) can be as arbitrarily small as the approximation order parameter n→∞ . To illustrate, we confirm that the assumptions are satisfied by two typical concrete models, a Bayesian linear inverse problem and a two-component mixture of Gaussian distributions.