skipping sampler

“The Skipping Sampler is an adaptation of the MH algorithm designed to sample from targets which have areas of zero density. It ‘skips’ across such areas, much as a flat stone can skip or skim repeatedly across the surface of water.”

An interesting challenge is simulating from a density restricted to a set C when little is known about C, apart from a mean to check whether or not a given value x is in C or not. John Moriarty, Jure Vogrinc (University of Warwick), and Alessandro Zocca make a new proposal to address this problem in a recently arXived paper. Which somewhat reminded me of the delayed rejection methods proposed by Antonietta Mira. And of our pinball sampler.

The paper spends a large amount of space about transferring from the Euclidean representation of the symmetric proposal density q to its polar representation. Which is rather trivial, but brings the questions of the efficient polar proposals  and of selecting the right type of Euclidean distance for the intended target. The method proposed therein is to select a direction first and keep skipping step by step in that direction until the set C is met again (re-entered). Or until a stopping (halting) boundary has been hit. This makes for a more complex proposal than usual but somewhat surprisingly the symmetry in q is sufficient to make the acceptance probability only depend on the target density.

While the convergence is properly established, I wonder at the practicality of the approach when compared with a regular random walk Metropolis algorithm in that both require a scaling to the jump that relates to the support of the target. Neither too small nor too large. If the set C is that unknown that only local (in or out) information is available, scaling of the jumps (and of the stopping rule) may prove problematic. In equivalent ways for both samplers. In a completely blind exploration, sequential (or population) Monte Carlo would seem more appropriate, at least to learn about the scale of jumps and location of the set C. If this set is defined as an intersection of constraints, a tempered (and sequential) solution would be helpful.  When checking the appurtenance to C becomes a computational challenge, more advance schemes have to be constructed, I would think.

dynamic nested sampling for stars

In the sequel of earlier nested sampling packages, like MultiNest, Joshua Speagle has written a new package called dynesty that manages dynamic nested sampling, primarily intended for astronomical applications. Which is the field where nested sampling is the most popular. One of the first remarks in the paper is that nested sampling can be more easily implemented by using a Uniform reparameterisation of the prior, that is, a reparameterisation that turns the prior into a Uniform over the unit hypercube. Which means in fine that the prior distribution can be generated from a fixed vector of uniforms and known transforms. Maybe not such an issue given that this is the prior after all.  The author considers this makes sampling under the likelihood constraint a much simpler problem but it all depends in the end on the concentration of the likelihood within the unit hypercube. And on the ability to reach the higher likelihood slices. I did not see any special trick when looking at the documentation, but reflected on the fundamental connection between nested sampling and this ability. As in the original proposal by John Skilling (2006), the slice volumes are “estimated” by simulated Beta order statistics, with no connection with the actual sequence of simulation or the problem at hand. We did point out our incomprehension for such a scheme in our Biometrika paper with Nicolas Chopin. As in earlier versions, the algorithm attempts at visualising the slices by different bounding techniques, before proceeding to explore the bounded regions by several exploration algorithms, including HMC.

“As with any sampling method, we strongly advocate that Nested Sampling should not be viewed as being strictly“better” or “worse” than MCMC, but rather as a tool that can be more or less useful in certain problems. There is no “One True Method to Rule Them All”, even though it can be tempting to look for one.”

When introducing the dynamic version, the author lists three drawbacks for the static (original) version. One is the reliance on this transform of a Uniform vector over an hypercube. Another one is that the overall runtime is highly sensitive to the choice the prior. (If simulating from the prior rather than an importance function, as suggested in our paper.) A third one is the issue that nested sampling is impervious to the final goal, evidence approximation versus posterior simulation, i.e., uses a constant rate of prior integration. The dynamic version simply modifies the number of point simulated in each slice. According to the (relative) increase in evidence provided by the current slice, estimated through iterations. This makes nested sampling a sort of inversted Wang-Landau since it sharpens the difference between slices. (The dynamic aspects for estimating the volumes of the slices and the stopping rule may hinder convergence in unclear ways, which is not discussed by the paper.) Among the many examples produced in the paper, a 200 dimension Normal target, which is an interesting object for posterior simulation in that most of the posterior mass rests on a ring away from the maximum of the likelihood. But does not seem to merit a mention in the discussion. Another example of heterogeneous regression favourably compares dynesty with MCMC in terms of ESS (but fails to include an HMC version).

[Breaking News: Although I wrote this post before the exciting first image of the black hole in M87 was made public and hence before I was aware of it, the associated AJL paper points out relying on dynesty for comparing several physical models of the phenomenon by nested sampling.]

 

slice sampling for Dirichlet mixture process

When working with my PhD student Changye in Dauphine this morning I realised that slice sampling also applies to discrete support distributions and could even be of use in such settings. That it works is (now) straightforward in that the missing variable representation behind the slice sampler also applies to densities defined with respect to a discrete measure. That this is useful transpires from the short paper of Stephen Walker (2007) where we saw this, as Stephen relies on the slice sampler to sample from the Dirichlet mixture model by eliminating the tail problem associated with this distribution. (This paper appeared in Communications in Statistics and it is through Pati & Dunson (2014) taking advantage of this trick that Changye found about its very existence. I may have known about it in an earlier life, but I had clearly forgotten everything!)

While the prior distribution (of the weights) of the Dirichlet mixture process is easy to generate via the stick breaking representation, the posterior distribution is trickier as the weights are multiplied by the values of the sampling distribution (likelihood) at the corresponding parameter values and they cannot be normalised. Introducing a uniform to replace all weights in the mixture with an indicator that the uniform is less than those weights corresponds to a (latent variable) completion [or a demarginalisation as we called this trick in Monte Carlo Statistical Methods]. As elaborated in the paper, the Gibbs steps corresponding to this completion are easy to implement, involving only a finite number of components. Meaning the allocation to a component of the mixture can be operated rather efficiently. Or not when considering that the weights in the Dirichlet mixture are not monotone, hence that a large number of them may need to be computed before picking the next index in the mixture when the uniform draw happens to be quite small.

common derivation for Metropolis–Hastings and other MCMC algorithms

Khoa Tran and Robert Kohn from UNSW just arXived a paper on a comprehensive derivation of a large range of MCMC algorithms, beyond Metropolis-Hastings. The idea is to decompose the MCMC move into

1. a random completion of the current value θ into V;
2. a deterministic move T from (θ,V) to (ξ,W), where only ξ matters.

If this sounds like a new version of Peter Green’s completion at the core of his 1995 RJMCMC algorithm, it is because it is indeed essentially the same notion. The resort to this completion allows for a standard form of the Metropolis-Hastings algorithm, which leads to the correct stationary distribution if T is self-inverse. This representation covers Metropolis-Hastings algorithms, Gibbs sampling, Metropolis-within-Gibbs and auxiliary variables methods, slice sampling, recursive proposals, directional sampling, Langevin and Hamiltonian Monte Carlo, NUTS sampling, pseudo-marginal Metropolis-Hastings algorithms, and pseudo-marginal Hamiltonian  Monte Carlo, as discussed by the authors. Given this representation of the Markov chain through a random transform, I wonder if Peter Glynn’s trick mentioned in the previous post on retrospective Monte Carlo applies in this generic setting (as it could considerably improve convergence…)

AISTATS 2016 [#2]

The second and third days of AISTATS 2016 passed like a blur, with not even the opportunity to write my impressions in real time! Maybe long tapa breaks are mostly to blame for this… In any case, we had two further exciting plenary talks about privacy-preserving data analysis by Kamalika Chaudhuri and crowdsourcing and machine learning by Adam Tauman Kalai. The talk by Kamalika was covering recent results by Kamalika and coauthors about optimal privacy preservation in classification and a generalisation to correlated data, with the neat notion of a Markov Quilt.  Other talks that same day also dwelt on this privacy issue, but I could not be . The talk by Adam was full of fun illustrations on humans training learning systems (with the unsolved difficulty of those humans deliberately mis-training the system, as exhibited recently by the short-lived Microsoft Tay experiment).

Both poster sessions were equally exciting, with the addition of MLSS student posters on the final day. Among many, I particularly enjoyed Iain Murray’s pseudo-marginal slice sampling, David Duvenaud’s fairly intriguing use of early stopping for non-parametric inference,  Garrett Bernstein’s work on aggregated Markov chains, Ye Wang’s scalable geometric density estimation [with a special bonus for his typo on the University of Turing, instead of Torino], Gemma Moran’s and Chengtao Li’s posters on determinantal processes, and Matej Balog’s Mondrian forests with a Laplace kernel [envisioning potential applications for ABC]. Again, just to mention a few…

The participants [incl. myself] also took one evening off to visit a sherry winery in Jerez, with a well-practiced spiel on the story of the company, with some building designed by Gutave Eiffel, and with a wine-tasting session. As I personally find this type of brandy too strong in alcohol, I am not a big fan of sherry but it was nonetheless an amusing trip! With no visible after-effects the next morning, since the audience was as large as usual for Adam’s talk [although I did not cross a machine-learning soul on my 6am run…]

In short, I enjoyed very much AISTATS 2016 and remain deeply impressed by the efficiency of the selection process and the amount of involvement of the actors of this selection, as mentioned earlier on the ‘Og. Kudos!