the buzz about nuzz

“…expensive in these terms, as for each root, Λ(x(s),v) (at the cost of one epoch) has to be evaluated for each root finding iteration, for each node of the numerical integral“

When using the ZigZag sampler, the main (?) difficulty is in producing velocity switch as the switches are produced as interarrival times of an inhomogeneous Poisson process. When the rate of this process cannot be integrated out in an analytical manner, the only generic approach I know is in using Poisson thinning, obtained by finding an integrable upper bound on this rate, generating from this new process and subsampling. Finding the bound is however far from straightforward and may anyway result in an inefficient sampler. This new paper by Simon Cotter, Thomas House and Filippo Pagani makes several proposals to simplify this simulation, Nuzz standing for numerical ZigZag. Even better (!), their approach is based on what they call the Sellke construction, with Tom Sellke being a probabilist and statistician at Purdue University (trivia: whom I met when spending a postdoctoral year there in 1987-1988) who also wrote a fundamental paper on the opposition between Bayes factors and p-values with Jim Berger.

“We chose as a measure of algorithm performance the largest Kolmogorov-Smirnov (KS) distance between the MCMC sample and true distribution amongst all the marginal distributions.”

The practical trick is rather straightforward in that it sums up as the exponentiation of the inverse cdf method, completed with a numerical resolution of the inversion. Based on the QAGS (Quadrature Adaptive Gauss-Kronrod Singularities) integration routine. In order to save time Kingman’s superposition trick only requires one inversion rather than d, the dimension of the variable of interest. This nuzzled version of ZIgZag can furthermore be interpreted as a PDMP per se. Except that it retains a numerical error, whose impact on convergence is analysed in the paper. In terms of Wasserstein distance between the invariant measures. The paper concludes with a numerical comparison between Nuzz and random walk Metropolis-Hastings, HMC, and manifold MALA, using the number of evaluations of the likelihood as a measure of time requirement. Tuning for Nuzz is described, but not for the competition. Rather dramatically the Nuzz algorithm performs worse than this competition when counting one epoch for each likelihood computation and better when counting one epoch for each integral inversion. Which amounts to perfect inversion, unsurprisingly. As a final remark, all models are more or less Normal, with very smooth level sets, maybe not an ideal range

 

crepuscule with [N]ellie[s]

ABC World seminar

With most of the World being more or less confined at home, conferences cancelled one after the other, including ABC in Grenoble!, we are launching a fortnightly webinar on approximation Bayesian computation, methods, and inference. The idea is to gather members and disseminate results and innovation during these coming weeks and months under lock-down. And hopefully after!

At this point, the interface will be Blackboard Collaborate, run from Edinburgh by Michael Gutmann, for which neither registration nor software is required. Before each talk, a guest link will be mailed to the mailing list. Please register here to join the list.

The seminar is planned on Thursdays at either 9am or more likely 11:30 am UK (+1GMT) time, as we are still debating the best schedule to reach as many populated time zones as possible!, and the first speakers are

09.04.2020	Dennis Prangle	Distilling importance sampling
23.04.2020	Ivis Kerama and Richard Everitt	Rare event SMC²
07.05.2020	Umberto Picchini	Stratified sampling and bootstrapping for ABC

COMPSTAT 2020 moved to 2021

Just received the news that the COMPSTAT 2020 meeting that was supposed to take place in Bologna, late August 2020, has been postponed by a year. Meaning that, reasonably, all future COMPSTAT conferences are postponed by a year. This gap policy should apply to all conference cycles, I believe.

nested sampling via SMC

“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”

A paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?

1. Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
2. The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
3. There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.

Not mentioning the Monte Carlo error, of course, which should remain at the √n level.

“Nested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”

The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper.  (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.

“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”

A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side,  I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)

misspecified [but published!]

BFF⁷ postponed