Hamiltonian Monte Carlo, doubly intractable distributions and other challenges

Jeudi 17 janvier 2019 14:00-15:00 - Julien Stoehr - CEREMADE - Université Paris Dauphine

Résumé : 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. I will illustrate this on a Potts model example and an ERGM example. The discretised flow also requires some amount of tuning and calibration. I will present an essentially calibration free version of the algorithm based on the distribution of the integration time of the associated integrator. When compared with the original NUTS (golden standard) on benchmarks, this algorithm exhibits a significantly improved efficiency.

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