Speaker: Bob Carpenter (Flatiron Institute)

Session Chair: Sam Livingstone (University College London)

Abstract: In this talk, I will demonstrate how we can combine two ideas, generalized Hamiltonian Monte Carlo and delayed rejection, to derive a sampler that is as efficient as Hamiltonian Monte Carlo, but is able to adapt its step size to deal with multiscale distributions, much like a standard integrator for ordinary differential equations. A distribution is multiscale if its curvature has different scales in the posterior; a textbook example is Radford Neal’s funnel example derived from hierarchical priors, which has a very flat mouth (corresponding to high population variance) and very highly curved neck (low population variance). No fixed step size allows exploration of its posterior. Generalized HMC allows us to take a single Hamiltonian step along the gradient at a time (which is equivalent to Metropolis-adjusted Langevin dynamics), but only refresh momentum partially (which makes it underdamped). The naive form of this algorithm does not work because momentum must be reversed to maintain detailed balance if the Metropolis step rejects. To maintain directed exploration, we apply delayed rejection, which allows a proposal rejected due to divergence of the Hamiltonian (from too large a step size in the first-order approximation of the dynamics) to be retried with a smaller step size (with a Hastings-style adjustment for the retry). We show that the combination of delayed rejection and GHMC allows sampling multiscale distributions which otherwise lead to biased samples in standard Hamiltonian Monte Carlo (including dynamic forms such as the no-U-turn sampler). In conclusion, I will discuss some preliminary work on applying the the automatic tuning method using complementary parallel chains developed by Matt Hoffman and Pavel Sountsov for their sampler MEADS (which also uses generalized HMC, but with an alternative approach to maintaining directed exploration based on work of Radford neal, which will also describe).