population quasi-Monte Carlo

“Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to generate weighted samples that approximate the target distribution”

A return of the prodigal son!, with this arXival by Huang, Joseph, and Mak, of a paper on population Monte Carlo using quasi-random sequences. The construct is based on an earlier notion of Joseph and Mak, support points, which are defined wrt a given target distribution F as minimising the variability of a sample from F away from these points. (I would have used instead my late friend Bernhard Flury’s principal points!) The proposal uses Owen-style scrambled Sobol points, followed by a deterministic mixture weighting à la PMC, followed by importance support resampling to find the next location parameters of the proposal mixture (which is why I included an unrelated mixture surface as my post picture!). This importance support resampling is obviously less variable than the more traditional ways of resampling but the cost moves from O(M) to O(M²).

“The main computational complexity of the algorithm is O(M²) from computing the pairwise distance of the M weighted samples”

The covariance parameters are updated as in our 2008 paper. This new proposal is interesting and reasonable, with apparent significant gains, albeit I would have liked to see a clearer discussion of the actual computing costs of PQMC.

ABC in Clermont-Ferrand

Today I am taking part in a one-day workshop at the Université of Clermont Auvergne on ABC. With applications to cosmostatistics, along with Martin Kilbinger [with whom I worked on PMC schemes], Florent Leclerc and Grégoire Aufort. This should prove a most exciting day! (With not enough time to run up Puy de Dôme in the morning, though.)

SMC on a sequence of increasing dimension targets

Richard Everitt and co-authors have arXived a preliminary version of a paper entitled Sequential Bayesian inference for mixture models and the coalescent using sequential Monte Carlo samplers with transformations. The central notion is an SMC version of the Carlin & Chib (1995) completion in the comparison of models in different dimensions. Namely to create auxiliary variables for each model in such a way that the dimension of the completed models are all the same. (Reversible jump MCMC à la Peter Green (1995) can also be interpreted this way, even though only relevant bits of the completion are used in the transitions.) I find the paper and the topic most interesting if only because it relates to earlier papers of us on population Monte Carlo. It also brought to my awareness the paper by Karagiannis and Andrieu (2013) on annealed reversible jump MCMC that I had missed at the time it appeared. The current paper exploits this annealed expansion in the devising of the moves. (Sequential Monte Carlo on a sequence of models with increasing dimension has been studied in the past.)

The way the SMC is described in the paper, namely, reweight-subsample-move, does not strike me as the most efficient as I would try to instead move-reweight-subsample, using a relevant move that incorporate the new model and hence enhance the chances of not rejecting.

One central application of the paper is mixture models with an unknown number of components. The SMC approach applied to this problem means creating a new component at each iteration t and moving the existing particles after adding the parameters of the new component. Since using the prior for this new part is unlikely to be at all efficient, a split move as in Richardson and Green (1997) can be considered, which brings back the dreaded Jacobian of RJMCMC into the picture! Here comes an interesting caveat of the method, namely that the split move forces a choice of the split component of the mixture. However, this does not appear as a strong difficulty, solved in the paper by auxiliary [index] variables, but possibly better solved by a mixture representation of the proposal, as in our PMC [population Monte Carlo] papers. Which also develop a family of SMC algorithms, incidentally. We found there that using a mixture representation of the proposal achieves a provable variance reduction.

“This puts a requirement on TSMC that the single transition it makes must be successful.”

As pointed by the authors, the transformation SMC they develop faces the drawback that a given model is only explored once in the algorithm, when moving to the next model. On principle, there would be nothing wrong in including regret steps, retracing earlier models in the light of the current one, since each step is an importance sampling step valid on its own right. But SMC also offers a natural albeit potentially high-varianced approximation to the marginal likelihood, which is quite appealing when comparing with an MCMC outcome. However, it would have been nice to see a comparison with alternative estimates of the marginal in the case of mixtures of distributions. I also wonder at the comparative performances of a dual approach that would be sequential in the number of observations as well, as in Chopin (2004) or our first population Monte Carlo paper (Cappé et al., 2005), since subsamples lead to tempered versions of the target and hence facilitate moves between models, being associated with flatter likelihoods.

PMC for combinatoric spaces

I received this interesting [edited] email from Xiannian Fan at CUNY:

I am trying to use PMC to solve Bayesian network structure learning problem (which is in a combinatorial space, not continuous space).

In PMC, the proposal distributions q_i,t can be very flexible, even specific to each iteration and each instance. My problem occurs due to the combinatorial space.

For importance sampling, the requirement for proposal distribution, q, is:

support (p) ⊂ support (q)             (*)

For PMC, what is the support of the proposal distribution in iteration t? is it

support (p) ⊂ U support(q_i,t)    (**)

or does (*) apply to every q_i,t?

For continuous problem, this is not a big issue. We can use random walk of Normal distribution to do local move satisfying (*). But for combination search, local moving only result in finite states choice, just not satisfying (*). For example for a permutation (1,3,2,4), random swap has only choose(4,2)=6 neighbor states.

Fairly interesting question about population Monte Carlo (PMC), a sequential version of importance sampling we worked on with French colleagues in the early 2000’s.  (The name population Monte Carlo comes from Iba, 2000.)  While MCMC samplers do not have to cover the whole support of p at each iteration, it is much harder for importance samplers as their core justification is to provide an unbiased estimator to for all integrals of interest. Thus, when using the PMC estimate,

1/n ∑_i,t {p(x_i,t)/q_i,t(x_i,t)}h(q_i,t),  x_i,t~q_i,t(x)

this estimator is only unbiased when the supports of the q_i,t “s are all containing the support of p. The only other cases I can think of are

1. associating the q_i,t “s with a partition S_i,t of the support of p and using instead

   ∑_i,t {p(x_i,t)/q_i,t(x_i,t)}h(q_i,t), x_i,t~q_i,t(x)

2. resorting to AMIS under the assumption (**) and using instead

   1/n ∑_i,t {p(x_i,t)/∑_j,t q_j,t(x_i,t)}h(q_i,t), x_i,t~q_i,t(x)

but I am open to further suggestions!

Initializing adaptive importance sampling with Markov chains

Another paper recently arXived by Beaujean and Caldwell elaborated on our population Monte Carlo papers (Cappé et al., 2005, Douc et al., 2007, Wraith et al., 2010) to design a more thorough starting distribution. Interestingly, the authors mention the fact that PMC is an EM-type algorithm to emphasize the importance of the starting distribution, as with “poor proposal, PMC fails as proposal updates lead to a consecutively poorer approximation of the target” (p.2). I had not thought of this possible feature of PMC, which indeed proceeds along integrated EM steps, and thus could converge to a local optimum (if not poorer than the start as the Kullback-Leibler divergence decreases).

The solution proposed in this paper is similar to the one we developed in our AMIS paper. An important part of the simulation is dedicated to the construction of the starting distribution, which is a mixture deduced from multiple Metropolis-Hastings runs. I find the method spends an unnecessary long time on refining this mixture by culling the number of components: down-the-shelf clustering techniques should be sufficient, esp. if one considers that the value of the target is available at every simulated point. This has been my pet (if idle) theory for a long while: we do not take (enough) advantage of this informative feature in our simulation methods… I also find the Student’s t versus Gaussian kernel debate (p.6) somehow superfluous: as we shown in Douc et al., 2007, we can process Student’s t distributions so we can as well work with those. And rather worry about the homogeneity assumption this choice implies: working with any elliptically symmetric kernel assumes a local Euclidean structure on the parameter space, for all components, and does not model properly highly curved spaces. Another pet theory of mine’s. As for picking the necessary number of simulations at each PMC iteration, I would add to the ESS and the survival rate of the components a measure of the Kullback-Leibler divergence, as it should decrease at each iteration (with an infinite number of particles).

Another interesting feature is in the comparison with Multinest, the current version of nested sampling, developed by Farhan Feroz. This is the second time I read a paper involving nested sampling in the past two days. While this PMC implementation does better than nested sampling on the examples processed in the paper, the Multinest outcome remains relevant, particularly because it handles multi-modality fairly well. The authors seem to think parallelisation is an issue with nested sampling, while I do see why: at the most naïve stage, several nested samplers can be run in parallel and the outcomes pulled together.