non-reversible guided Metropolis–Hastings

Kengo Kamatani and Xiaolin Song, whom I visited in Osaka last summer in what seems like another reality!, just arXived another paper on a non-reversible Metropolis version. That exploits a group action and the associated Haar measure.

Following a proposal of Gustafson (1998), a ∆-guided Metropolis–Hastings kernel is based on a statistic ∆ that is totally ordered and determine the acceptance of a proposed value y~Q(x,.) by adding a direction (-,+) to the state space and moving from x if ∆x≤∆y in the positive direction and if ∆y≤∆x in the negative direction [with the standard Metropolis–Hastings acceptance probability]. The sign of the direction switches in case of a rejection. And the statistic ∆ is such that the proposal kernel Q(x,.) is unbiased, i.e., agnostic to the sign, i.e., it gives the same probability to ∆x≤∆y and ∆y≤∆x. This modification reduces the asymptotic variance compared with the original Metropolis–Hastings kernel.

To construct a random walk proposal that is unbiased, the authors assume that the ∆ transform takes values in a topological group, G, with Q further being invariant under the group actions. This can be constructed from a standard proposal by averaging the transforms of Q under all elements of the group over the associated right Haar measure. (Which I thought implied that the group is compact, except I forgot to account for the data update into a posterior..!) The worked-out example is based on a multivariate autoregressive kernel with ∆x being a rescaled non-central chi-squared variate. In dimension 24. The results show a clear improvement in effective sample size per second evaluation over off-the-shelf random walk and Hamiltonian Monte Carlo versions.

Seeing the Haar measure appearing in the setting of Markov chain Monte Carlo is fun!, as my last brush with it was not algorithmic. I would think the proposal only applies to settings where the components of the simulated vector are somewhat homogeneous in that the determinationthe determination of both the group action and a guiding statistic seem harder in cases where these components take different meaning (or live in a weird topology). I also lazily wonder if selecting the guiding statistic as a gradient of the log-target would have any interest.

coordinate sampler on-line

BayesComp’20

First, I really have to congratulate my friend Jim Hobert for a great organisation of the meeting adopting my favourite minimalist principles (no name tag, no “goodies” apart from the conference schedule, no official talks). Without any pretense at objectivity, I also appreciated very much the range of topics and the sweet frustration of having to choose between two or three sessions each time. Here are some notes taken during some talks (with no implicit implication for the talks no mentioned, re. above frustration! as well as very short nights making sudden lapse in concentration highly likely).

On Day 1, Paul Fearnhead’s inaugural plenary talk was on continuous time Monte Carlo methods, mostly bouncy particle and zig-zag samplers, with a detailed explanation on the simulation of the switching times which likely brought the audience up to speed even if they had never heard of them. And an opening on PDMPs used as equivalents to reversible jump MCMC, reminding me of the continuous time (point process) solutions of Matthew Stephens for mixture inference (and of Preston, Ripley, Møller).

The same morn I heard of highly efficient techniques to handle very large matrices and p>n variables selections by Akihiko Nishimura and Ruth Baker on a delayed acceptance ABC, using a cheap proxy model. Somewhat different from indirect inference. I found the reliance on ESS somewhat puzzling given the intractability of the likelihood (and the low reliability of the frequency estimate) and the lack of connection with the “real” posterior. At the same ABC session, Umberto Picchini spoke on a joint work with Richard Everitt (Warwick) on linking ABC and pseudo-marginal MCMC by bootstrap. Actually, the notion of ABC likelihood was already proposed as pseudo-marginal ABC by Anthony Lee, Christophe Andrieu and Arnaud Doucet in the discussion of Fearnhead and Prangle (2012) but I wonder at the focus of being unbiased when the quantity is not the truth, i.e. the “real” likelihood. It would seem more appropriate to attempt better kernel estimates on the distribution of the summary itself. The same session also involved David Frazier who linked our work on ABC for misspecified models and an on-going investigation of synthetic likelihood.

Later, there was a surprise occurrence of the Bernoulli factory in a talk by Radu Herbei on Gaussian process priors with accept-reject algorithms, leading to exact MCMC, although the computing implementation remains uncertain. And several discussions during the poster session, incl. one on the planning of a 2021 workshop in Oaxaca centred on objective Bayes advances as we received acceptance of our proposal by BIRS today!

On Day 2, David Blei gave a plenary introduction to variational Bayes inference and latent Dirichlet allocations, somewhat too introductory for my taste although other participants enjoyed this exposition. He also mentioned a recent JASA paper on the frequentist consistency of variational Bayes that I should check. Speaking later with PhD students, they really enjoyed this opening on an area they did not know that well.

A talk by Kengo Kamatani (whom I visited last summer) on improved ergodicity rates for heavy tailed targets and Crank-NIcholson modifications to the random walk proposal (which uses an AR(1) representation instead of the random walk). With the clever idea of adding the scale of the proposal as an extra parameter with a prior of its own. Gaining one order of magnitude in the convergence speed (i.e. from d to 1 and from d² to d, where d is the dimension), which is quite impressive (and just published in JAP).Veronica Rockova linked Bayesian variable selection and machine learning via ABC, with conditions on the prior for model consistency. And a novel approach using part of the data to learn an ABC partial posterior, which reminded me of the partial  Bayes factors of the 1990’s although it is presumably unrelated. And a replacement of the original rejection ABC via multi-armed bandits, where each variable is represented by an arm, called ABC Bayesian forests. Recalling the simulation trick behind Thompson’s approach, reproduced for the inclusion or exclusion of variates and producing a fixed estimate for the (marginal) inclusion probabilities, which makes it sound like a prior-feeback form of empirical Bayes. Followed by a talk of Gregor Kastner on MCMC handling of large time series with specific priors and a massive number of parameters.

The afternoon also had a wealth of exciting talks and missed opportunities (in the other sessions!). Which ended up with a strong if unintended French bias since I listened to Christophe Andrieu, Gabriel Stolz, Umut Simsekli, and Manon Michel on different continuous time processes, with Umut linking GANs, multidimensional optimal transport, sliced-Wasserstein, generative models, and new stochastic differential equations. Manon Michel gave a highly intuitive talk on creating non-reversibility, getting rid of refreshment rates in PDMPs to kill any form of reversibility.

non-reversibility in discrete spaces

Following a recent JASA paper by Giacomo Zanella (which I have not yet read but is discussed on this blog), Sam Power and Jacob Goldman have recently arXived a paper on Accelerated sampling on discrete spaces with non-reversible Markov processes, where they use continuous-time, non-reversible algorithms à la PDMP, even though differential equations do not exist on discrete spaces. More specifically, they devise discrete versions of the coordinate sampler and of the Zig-Zag sampler, using Markov jump processes instead of differential equations, with detailed balance on the jump rate rather than the Markov kernel. A use of jump processes originating at least from Peskun (1973) and connected with MCMC algorithms in Matthew Stephens‘ 1999 PhD thesis. A neat thing about discrete settings is that the jump process can be implemented with no discretisation! However, as we noticed when working on birth-and-death processes with Olivier Cappé and Tobias Rydèn, there is a potential for disastrous implementation if an infinite sequence of instantaneous moves (out of zero probability states) is proposed.

The authors make the further assumption(s) that the discrete space is endowed with a graphical structure with a group G acting upon this graph, with an involution keeping the target (or a completion of the original target) invariant. In this framework, reversibility amounts to repeatedly using (group) generators þ with a low order (as in Bayesian variable selection, binary spin systems, where þ.þ=id, and other permutation problems), since they bring the chain back to its starting point. Their first sampler is called a Tabu sampler for avoiding such behaviour, forcing the next step to use other generators þ in the generator set Þ thanks to a binary auxiliary variable that partitions Þ into forward vs backward moves. For high order generators, the discrete coordinate and Zig-Zag samplers are instead repeatedly using the same generator (although it is unclear to me why this is beneficial, given that neither graph nor generator is not necessarily linked with the target). With the coordinate sampler being again much cheaper since it only looks at one direction in the generator group.

The paper contains a range of comparisons with (only) Zanella’s sampler, some presenting heavy gains in terms of ESS. Including one on hundreds of sensors in a football stadium. As I am not particularly familiar with these examples, except for the Bayesian variable selection one, I found it rather hard to determine whether or not the compared samplers were indeed exploring the entirety of the (highly complex and highly dimensional) target. The collection of examples is however quite rich and support the use of such non-reversible schemes. It may also be that the discrete nature of the target could facilitate the theoretical study of their convergence properties.

BayesComp 2020 at a glance