Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within thestatisticians toolbox as an alternative sampling method in settings when standardMetropolis-Hastings is inefficient. HMC generates a Markov chain on an augmentedstate space with transitions based on a deterministic differential flow derived fromHamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannotbe solved analytically, requiring numerical integration schemes. Under numericalintegration, the resulting approximate solution no longer preserves the measure ofthe target distribution, therefore an accept-reject step is used to correct the bias.For doubly-intractable distributions such as posterior distributions based on Gibbsrandom fields HMC suffers from some computational difficulties: computationof gradients in the differential flow and computation of the accept-reject proposalsposes difficulty. In this paper, we study the behaviour of HMC when these quantitiesare replaced by Monte Carlo estimates.