Noisy Hamiltonian Monte Carlo for doubly intractable distributions


Seminar Probabilités & Statistique

10/01/2019 - 14:00 Mr Julien Stoehr (Université Paris-Dauphine) Salle 106 - Batiment IMAG

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician's toolbox as an alternative sampling method in settings when standard Metropolis-Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics.  In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept-reject step is used to correct the bias. For doubly-intractable distributions -- such as posterior distributions based on Gibbs random fields (e.g., Potts model, ERGM), HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept-reject proposals poses difficulty. In this talk, I will present the behaviour of HMC when these quantities are replaced by Monte Carlo estimates and discuss briefly calibration issues. I will illustrate this on a Potts model example and an ERGM example.