## extending ABC to high dimensions via Gaussian copula

Posted in Books, pictures, Statistics, Travel, Uncategorized, University life with tags , , , on April 28, 2015 by xi'an

Li, Nott, Fan, and Sisson arXived last week a new paper on ABC methodology that I read on my way to Warwick this morning. The central idea in the paper is (i) to estimate marginal posterior densities for the components of the model parameter by non-parametric means; and (ii) to consider all pairs of components to deduce the correlation matrix R of the Gaussian (pdf) transform of the pairwise rank statistic. From those two low-dimensional estimates, the authors derive a joint Gaussian-copula distribution by using inverse  pdf transforms and the correlation matrix R, to end up with a meta-Gaussian representation

$f(\theta)=\dfrac{1}{|R|^{1/2}}\exp\{\eta^\prime(I-R^{-1})\eta/2\}\prod_{i=1}^p g_i(\theta_i)$

where the η’s are the Gaussian transforms of the inverse-cdf transforms of the θ’s,that is,

$\eta_i=\Phi^{-1}(G_i(\theta_i))$

Or rather

$\eta_i=\Phi^{-1}(\hat{G}_i(\theta_i))$

given that the g’s are estimated.

This is obviously an approximation of the joint in that, even in the most favourable case when the g’s are perfectly estimated, and thus the components perfectly Gaussian, the joint is not necessarily Gaussian… But it sounds quite interesting, provided the cost of running all those transforms is not overwhelming. For instance, if the g’s are kernel density estimators, they involve sums of possibly a large number of terms.

One thing that bothers me in the approach, albeit mostly at a conceptual level for I realise the practical appeal is the use of different summary statistics for approximating different uni- and bi-dimensional marginals. This makes for an incoherent joint distribution, again at a conceptual level as I do not see immediate practical consequences… Those local summaries also have to be identified, component by component, which adds another level of computational cost to the approach, even when using a semi-automatic approach as in Fernhead and Prangle (2012). Although the whole algorithm relies on a single reference table.

The examples in the paper are (i) the banana shaped “Gaussian” distribution of Haario et al. (1999) that we used in our PMC papers, with a twist; and (ii) a g-and-k quantile distribution. The twist in the banana (!) is that the banana distribution is the prior associated with the mean of a Gaussian observation. In that case, the meta-Gaussian representation seems to hold almost perfectly, even in p=50 dimensions. (If I remember correctly, the hard part in analysing the banana distribution was reaching the tails, which are extremely elongated in at least one direction.) For the g-and-k quantile distribution, the same holds, even for a regular ABC. What seems to be of further interest would be to exhibit examples where the meta-Gaussian is clearly an approximation. If such cases exist.

## optimal mixture weights in multiple importance sampling

Posted in Statistics, University life with tags , , , , , , on December 12, 2014 by xi'an

Multiple importance sampling is back!!! I am always interested in this improvement upon regular importance sampling, even or especially after publishing a recent paper about our AMIS (for adaptive multiple importance sampling) algorithm, so I was quite eager to see what was in Hera He’s and Art Owen’s newly arXived paper. The paper is definitely exciting and set me on a new set of importance sampling improvements and experiments…

Some of the most interesting developments in the paper are that, (i) when using a collection of importance functions qi with the same target p, every ratio qi/p is a control variate function with expectation 1 [assuming each of the qi‘s has a support smaller than the support of p]; (ii) the weights of a mixture of the qi‘s can be chosen in an optimal way towards minimising the variance for a certain integrand; (iii) multiple importance sampling incorporates quite naturally stratified sampling, i.e. the qi‘s may have disjoint supports; )iv) control variates contribute little, esp. when compared with the optimisation over the weights [which does not surprise me that much, given that the control variates have little correlation with the integrands]; (v) Veach’s (1997) seminal PhD thesis remains a driving force behind those results [and in getting Eric Veach an Academy Oscar in 2014!].

One extension that I would find of the uttermost interest deals with unscaled densities, both for p and the qi‘s. In that case, the weights do not even sum up to a know value and I wonder at how much more difficult it is to analyse this realistic case. And unscaled densities led me to imagine using geometric mixtures instead. Or even harmonic mixtures! (Maybe not.)

Another one is more in tune with our adaptive multiple mixture paper. The paper works with regret, but one could also work with remorse! Besides the pun, this means that one could adapt the weights along iterations and even possible design new importance functions from the past outcome, i.e., be adaptive once again. He and Owen suggest mixing their approach with our adaptive sequential Monte Carlo model.

## efficient exploration of multi-modal posterior distributions

Posted in Books, Statistics, University life with tags , , , , on September 1, 2014 by xi'an

The title of this recent arXival had potential appeal, however the proposal ends up being rather straightforward and hence  anti-climactic! The paper by Hu, Hendry and Heng proposes to run a mixture of proposals centred at the various modes of  the target for an efficient exploration. This is a correct MCMC algorithm, granted!, but the requirement to know beforehand all the modes to be explored is self-defeating, since the major issue with MCMC is about modes that are  omitted from the exploration and remain undetected throughout the simulation… As provided, this is a standard MCMC algorithm with no adaptive feature and I would rather suggest our population Monte Carlo version, given the available information. Another connection with population Monte Carlo is that I think the performances would improve by Rao-Blackwellising the acceptance rate, i.e. removing the conditioning on the (ancillary) component of the index. For PMC we proved that using the mixture proposal in the ratio led to an ideally minimal variance estimate and I do not see why randomising the acceptance ratio in the current case would bring any improvement.

## PMC for combinatoric spaces

Posted in Statistics, University life with tags , , , , , , , on July 28, 2014 by xi'an

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 qi,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(qi,t)    (**)

or does (*) apply to every qi,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(xi,t)/qi,t(xi,t)}h(qi,t),  xi,t~qi,t(x)

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

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

i,t {p(xi,t)/qi,t(xi,t)}h(qi,t), xi,t~qi,t(x)

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

1/n ∑i,t {p(xi,t)/∑j,t qj,t(xi,t)}h(qi,t), xi,t~qi,t(x)

but I am open to further suggestions!

## ABC for design

Posted in Statistics with tags , , , , , , , on August 30, 2013 by xi'an

I wrote a comment on this arXived paper on simulation based design that starts from Müller (1999) and gets an ABC perspective a while ago on my iPad when travelling to Montpellier and then forgot to download it…

Hainy, [Wener] Müller, and Wagner recently arXived a paper called “Likelihood-free Simulation-based Optimal Design“, paper which relies on ABC to construct optimal designs . Remember that [Peter] Müller (1999) uses a natural simulated annealing that is quite similar to our MAP [SAME] algorithm with Arnaud Doucet and Simon Godsill, relying on multiple versions of the data set to get to the maximum. The paper also builds upon our 2006 JASA paper with my then PhD student Billy Amzal, Eric Parent, and Frederic Bois, paper that took advantage of the then emerging particle methods to improve upon a static horizon target. While our method is sequential in that it pursues a moving target, it does not rely on the generic methodology developed by del Moral et al. (2006), where a backward kernel brings more stability to the moves. The paper also implements a version of our population Monte Carlo ABC algorithm (Beaumont et al., 2009), as a first step before an MCMC simulation. Overall, the paper sounds more like a review than like a strongly directive entry into ABC based design in that it remains quite generic. Not that I have specific suggestions, mind!, but I fear a realistic implementation (as opposed to the linear model used in the paper) would require a certain amount of calibration. There are missing references of recent papers using ABC for design, including some by Michael Stumpf I think.

I did not know about the Kuck et al. reference… Which is reproducing our 2006 approach within the del Moral framework. It uses a continuous temperature scale that I find artificial and not that useful, again a maybe superficial comment as I didn’t get very much into the paper … Just that integer powers lead to multiples of the sample and have a nice algorithmic counterpart.

## Initializing adaptive importance sampling with Markov chains

Posted in Statistics with tags , , , , , , , , , , , on May 6, 2013 by xi'an

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.

## ABC-MCMC for parallel tempering

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on February 9, 2012 by xi'an

In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the Parallel Tempering algorithm (Geyer, 1991).

Another of those arXiv papers that had sat on my to-read pile for way too long: Likelihood-free parallel tempering by Meïli Baragtti, Agnès Grimaud, and Denys Pommeret, from Luminy, Marseilles. The paper mentions our population Monte Carlo (PMC) algorithm (Beaumont et al., 2009) and other ABC-SMC algorithms, but opts instead for an ABC-MCMC basis. The purpose is to build a parallel tempering method. Tolerances and temperatures evolve simultaneously. I however fail to see where the tempering occurs in the algorithm (page 7): there is a set of temperatures T1,….,TN, but they do not appear within the algorithm. My first idea of a tempering mechanism in a likelihood-free setting was to replicate our SAME algorithm (Doucet, Godsill, and Robert, 2004), by creating Tj copies of the [pseudo-]observations to mimic the likelihood taken to the power Tj. But this is annealing, not tempering, and I cannot think of the opposite of copies of the data. Unless of course a power of the likelihood can be simulated (and even then, what would the equivalent be for the data…?) Maybe a natural solution would be to operate some kind of data-attrition, e.g. by subsampling the original vector of observations.

Discussing the issue with Jean-Michel Marin, during a visit to Montpellier today, I realised that the true tempering came from the tolerances εi, while the temperatures Tj were there to calibrate the proposal distributions. And that the major innovation contained in the thesis (if not so clearly in the paper) was to boost exchanges between different tolerances, improving upon the regular ABC-MCMC sampler by an equi-energy move.