**J**ulien Stoehr (now in Dublin, soon to join us as a new faculty in Paris-Dauphine!), Alan Benson and Nial Friel (both at UCD) arXived last week a paper entitled Noisy HMC for doubly-intractable distributions. Which considers solutions for adapting Hamiltonian Monte Carlo to target densities that involve a missing constant. In the sense of our workshop last year in Warwick. And in the theme pursued by Nial in the past years. The notion is thus to tackle a density π(θ)∞exp(V(X|θ)/Z(θ) when Z(θ) is intractable. In that case the gradient of log Z(θ) can be estimated as the expectation of the gradient of V(X|θ) [as a standard exponential family identity]. And the ratio of the Z(θ)’s appearing in the Metropolis ratio can be derived by Iain Murray’s exchange algorithm, based on simulations from the sampling distribution attached to the parameter in the denominator.

The resulting algorithm proposed by the authors thus uses N simulations of auxiliary variables at each step *þ* of the leapfrog part, towards an approximation of the gradient term, plus another N simulations for approximating the ratio of the normalising constants Z(θ)/Z(θ’). While justified from an importance sampling perspective, this approximation is quite poor when θ and θ’ differ. A better solution [as shown in the paper] is to take advantage of all leapfrog steps and of associated auxiliary simulations to build a telescopic product of ratios where the parameter values θ and θ’ are much closer. The main difficulty is in drawing a comparison with the exchange algorithm, since the noisy HMC version is computationally more demanding. (A secondary difficulty is in having an approximate algorithm that no longer leaves the target density stationary.)