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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.”)

AMSI Lectures #8-10

This is the last series of lectures for my AMSI Lecture tour: in Brisbane, I gave the Rao-Blackwellisation talk this afternoon at the University of Queensland and will give tomorrow both the public lecture (for the second time) and the ABC for model choice at QUT. This will then see the end to this (almost) exhaustive if exhausting tour of Eastern Australian university towns… Brisbane has this great feature of connecting QUT with UQ by a fast boat, which meant I could work with Kerrie Mengersen on the revision of our ABCel paper in the morning and be at UQ ½ an hour later. A good thing as there are only three days left till I leave Australia after this seven week visit… (I am afraid there is no free time left for sampling the Brisbane dumplings!)

Xi’an Australian Tour 2012

Here is my schedule (so far) for my Australian trip this summer/winter… Looking forward meeting loads of interesting people, problems and places!

Tour Schedule

Date	Host Institution	Venue	Time	Title
12 July	Australian Statistical Conference	Meeting Room 11	9:30 am	Approximate Bayesian Computation for model selection
13 July	University of Adelaide	TBC	TBC	TBC
16 July	University of NSW	Via AGR	2 pm	ABC methods for Bayesian model choice
17 July	University of Western Sydney	TBC	TBC	Rao-Blackwellisation of sampling schemes
26 July	University of Melbourne	Russell Love theatre, Richard Berry (Bldg 160)	2 pm	Approximate Bayesian computation (ABC): advances and limitations
26 July	AMSI Public Lecture	TBC	6 pm	Simulation as a universal tool for statistics
27 July	Monash University, Econometrics and Business Statistics seminar	TBC	2 pm	ABC methods for Bayesian model choice
14 August	Australian National University	Seminar Room G35, John Dedman (Bldg 27)	2 pm	Approximate Bayesian computation (ABC): advances and limitations
15 August	University of Wollongong	CSSM Meeting (Goulburn)		Rao-Blackwellisation of sampling schemes
20 August	University of Queensland	Room N201, Building 50	2 pm	Rao-Blackwellisation of sampling schemes
21 August	Queensland University of Technology	GP-Z1064 Gibson Room	TBC	ABC methods for Bayesian model choice
21 August	Queensland University of Technology	GP-Z1064 Gibson Room	TBC	Simulation as a universal tool for statistics