We had a wonderful time (and an unusually large audience) at the mostly Monte Carlo seminar last week as Pierre del Moral and Bob Carpenter both presented on exciting recent developments of theirs! Pierre talked about Kantorovich contraction of Markov semigroups, which sounds rather daunting!, but actually covers fairly general and generic convergence results, using tools like potentials and Lyapunov contractions, reminding me of the early days of MCMC and the papers of Gareth Roberts (University of Warwick), Jeff Rosenthal, Richard Tweedie and others.
Bob then spoke about the latest version of NUTS, the within-orbit adaptive NUTS (WALNUTS) sampler, which adapts the step size at every leapfrog step in order to conserve the Hamiltonian and keep the path stable enough. The adaptation is facilitated by incorporating this step size as an extra parameter with an attached distribution, that the authors call Gibbs self tuning (GIST), for coupling tuning parameters and conditionally Gibbs-sampling them per iteration in Hamiltonian Monte Carlo. This has been done in the past, incl. in some of my papers (e.g., Andrieu & Robert, 2004), but I could not cite a particular reference during the seminar.
Further light reflections that came to mind during Bob’s talk:
- with NUTS, if cycling is feasible in a finite time, we could wait for a second passage at the starting point and then get back halfway (with the difficulty of detecting this second passage)
- changing the kinetic matrix at each leapfrog jump is actually Riemannian HMC (and with cubic cost!)
- the doubling mechanism in both the original NUTS and in biased progressive NUTS is simulation wasting
- but so is (surprise, surprise!) finding adaptive mass matrices for WALNUTS at reasonable costs
