Archive for bouncy particle sampler

Bouncing bouncy particle papers

Posted in Books, pictures, Statistics, University life with tags , , , , on July 27, 2017 by xi'an

Yesterday, two papers on bouncy particle samplers simultaneously appeared on arXiv, arxiv:1707.05200 by Chris Sherlock and Alex Thiery, and arxiv:1707.05296 by Paul Vanetti, Alexandre Bouchard-Côté, George Deligiannidis, and Arnaud Doucet. As a coordinated move by both groups of authors who had met the weeks before at the Isaac Newton Institute in Cambridge.

The paper by Sherlock and Thiery, entitled a discrete bouncy particle sampler, considers a delayed rejection approach that only requires point-wise evaluations of the target density. The delay being into making a speed flip move after a proposal involving a flip in the speed and a drift in the variable of interest is rejected. To achieve guaranteed ergodicity, they add a random perturbation as in our recent paper, plus another perturbation based on a Brownian argument. Given that this is a discretised version of the continuous-time bouncy particle sampler, the discretisation step δ need be calibrated. The authors follow a rather circumvoluted argument to argue in favour of seeking a maximum number of reflections (for which I have obviously no intuition). Overall, I find it hard to assess how much of an advance this is, even when simulations support the notion of a geometric convergence.

“Our results provide a cautionary example that in certain high-dimensional scenarios, it is still preferable to perform refreshment even when randomized bounces are used.” Vanetti et al.

The paper by Paul Vanetti and co-authors has a much more ambitious scale in that it unifies most of the work done so far in this area and relates piecewise deterministic processes, Hamiltonian Monte Carlo, and discrete versions, containing on top fine convergence results. The main idea is to improve upon the existing deterministic methods by taking (more) into account the target density. Hence the use of a bouncy particle sampler associated with the Hamiltonian (as in HMC). This borrows from an earlier slice sampler idea of Iain Murray, Ryan Adams, and David McKay (AISTATS 2010), exploiting an exact Hamiltonian dynamics for an approximation to the true target to explore its support. Except that bouncing somewhat avoids the slice step. The [eight] discrete bouncy particle particle samplers derived from this framework are both correct against the targeted distribution and do not require the simulation of event times. The paper distinguishes between global and local versions, the later exploiting conditional independence properties in the (augmented) target. Which sounds like a version of multiple slice sampling.

forward event-chain Monte Carlo

Posted in Statistics, Travel, University life with tags on July 24, 2017 by xi'an

One of the authors of this paper contacted me to point out their results arXived last February [and revised last month] as being related to our bouncy particle paper arXived two weeks ago. And to an earlier paper by Michel et al. (2014) published in the Journal of Chemical Physics. (The authors actually happen to work quite nearby, on a suburban road I take every time I bike to Dauphine!) I think one reason we missed this paper in our literature survey is the use of a vocabulary taken from Physics rather than our Monte Carlo community, as in, e.g., using “event chain” instead of “bouncy particle”… The paper indeed contains schemes similar to ours, as did the on-going work by Chris Sherlock and co-authors Chris presented last week at the Isaac Newton Institute workshop on scalability. (Although I had troubles reading its physics style, in particular the justification for stationarity or “global balance” and the use of “infinitesimals”.)

“…we would like to find the optimal set of directions {e} necessary for the ergodicity and  allowing for an efficient exploration of the target distribution.”

The improvement sought is about improving the choice of the chain direction at each direction change. In order to avoid the random walk behaviour. The proposal is to favour directions close to the gradient of the log-likelihood, keeping the orthogonal to this gradient constant in direction (as in our paper) if not in scale. (As indicated above I have trouble understanding the ergodicity proof, if not the irreducibility. I also do not see how solving (11), which should be (12), is feasible in general. And why (29) amounts to simulating from (27)…)

generalised bouncy particle sampler

Posted in Statistics with tags on June 19, 2017 by xi'an

My PhD student, Wu Changye, just completed a paper on an extension of the bouncy particle sampler, to which he associated me as we had discussed the contents and especially the proofs together, although I did not participate in the redaction of the paper. The bouncy particle sampler belongs to the collection of piecewise deterministic samplers, which also contains the recent zig-zag sampler of Joris Bierkens, Paul Fearnhead, and Gareth Roberts (Warwick).  It was introduced by Alexandre Bouchard-Côté, Sebastian Vollmer (Warwick) and Arnaud Doucet, to appear in JASA, and uses one particle that is characterised by the pair (position, velocity), the velocity only changing at random times determined by the target density. The change is also deterministic. Which may lead to a lack of irreducibility in the process and is solved by an extra Poisson process in the original paper. In the generalisation imagined by Changye, the bounce becomes partly random in the direction orthogonal to the gradient. The stationarity results of Bouchard-Côté, Vollmer and Doucet then extend to this setting. As does the ability to subsample and compute faster versions of the likelihood function.