## multiple try Metropolis

Posted in Books, Statistics, University life with tags , , , , , , on February 18, 2016 by xi'an

Luca Martino and Francisco Louzada recently wrote a paper in Computational Statistics about some difficulties with the multiple try Metropolis algorithm. This version of Metropolis by Liu et al. (2000) makes several proposals in parallel and picks one among them by multinomial sampling where the weights are proportional to the corresponding importance weights. This is followed by a Metropolis acceptance step that requires simulating the same number of proposed moves from the selected value. While this is necessary to achieve detailed balance, this mixture of MCMC and importance sampling is inefficient in that it simulates a large number of particles and ends up using only one of them. By comparison, a particle filter for the same setting would propagate all N particles along iterations and only resamples occasionaly when the ESS is getting too small. (I also wonder if the method could be seen as a special kind of pseudo-marginal approach, given that the acceptance ratio is an empirical average with expectation the missing normalising constan [as I later realised the authors had pointed out!]… In which case efficiency comparisons by Christophe Andrieu and Matti Vihola could prove useful.)

The issue raised by Martino and Louzada is that the estimator of the normalising constant can be poor at times, especially when the chain is in low regions of the target, and hence get the chain stuck. The above graph illustrates this setting in the paper. However, the reason for the failure is mostly that the proposal distribution is inappropriate for the purpose of approximating the normalising constant, i.e., that importance sampling does not converge in this situation, since otherwise the average of the importance weights should a.s. converge to the normalising constant. And the method should not worsen when increasing the number of proposals at a given stage. (The solution proposed by the authors to have a random number of proposals seems unlikely to solve the issue in a generic situation. Changing the proposals towards different tail behaviours as in population Monte Carlo is more akin to defensive sampling and thus more likely to avoid trapping states. Interestingly, the authors eventually resort to a mixture denominator in the importance sampler following AMIS.)

## multiple importance sampling

Posted in Books, Statistics, University life with tags , , , , , , , , on November 20, 2015 by xi'an

“Within this unified context, it is possible to interpret that all the MIS algorithms draw samples from a equal-weighted mixture distribution obtained from the set of available proposal pdfs.”

In a very special (important?!) week for importance sampling!, Elvira et al. arXived a paper about generalized multiple importance sampling. The setting is the same as in earlier papers by Veach and Gibas (1995) or Owen and Zhou (2000) [and in our AMIS paper], namely a collection of importance functions and of simulations from those functions. However, there is no adaptivity for the construction of the importance functions and no Markov (MCMC) dependence on the generation of the simulations.

“One of the goals of this paper is to provide the practitioner with solid theoretical results about the superiority of some specific MIS schemes.”

One first part deals with the fact that a random point taken from the conjunction of those samples is distributed from the equiweighted mixture. Which was a fact I had much appreciated when reading Owen and Zhou (2000). From there, the authors discuss the various choices of importance weighting. Meaning the different degrees of Rao-Blackwellisation that can be applied to the sample. As we discovered in our population Monte Carlo research [which is well-referred within this paper], conditioning too much leads to useless adaptivity. Again a sort of epiphany for me, in that a whole family of importance functions could be used for the same target expectation and the very same simulated value: it all depends on the degree of conditioning employed for the construction of the importance function. To get around the annoying fact that self-normalised estimators are never unbiased, the authors borrow Liu’s (2000) notion of proper importance sampling estimators, where the ratio of the expectations is returning the right quantity. (Which amounts to recover the correct normalising constant(s), I believe.) They then introduce five (5!) different possible importance weights that all produce proper estimators. However, those weights correspond to different sampling schemes, so do not apply to the same sample. In other words, they are not recycling weights as in AMIS. And do not cover the adaptive cases where the weights and parameters of the different proposals change along iterations. Unsurprisingly, the smallest variance estimator is the one based on sampling without replacement and an importance weight made of the entire mixture. But this result does not apply for the self-normalised version, whose variance remains intractable.

I find this survey of existing and non-existing multiple importance methods quite relevant and a must-read for my students (and beyond!). My reservations (for reservations there must be!) are that the study stops short of pushing further the optimisation. Indeed, the available importance functions are not equivalent in terms of the target and hence weighting them equally is sub-efficient. The adaptive part of the paper broaches upon this issue but does not conclude.

## trip to München

Posted in Mountains, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , on October 19, 2015 by xi'an

While my train ride to the fabulous De Gaulle airport was so much delayed that I had less than ten minutes from jumping from the carriage to sitting in my plane seat, I handled the run through security and the endless corridors of the airport in the allotted time, and reached Munich in time for my afternoon seminar and several discussions that prolonged into a pleasant dinner of Wiener Schnitzel and Eisbier.  This was very exciting as I met physicists and astrophysicists involved in population Monte Carlo and parallel MCMC and manageable harmonic mean estimates and intractable ABC settings (because simulating the data takes eons!). I wish the afternoon could have been longer. And while this is the third time I come to Munich, I still have not managed to see the centre of town! Or even the nearby mountains. Maybe an unsuspected consequence of the Heisenberg principle…

Posted in Books, pictures, Statistics, University life with tags , , , , , , on July 30, 2015 by xi'an

Ingmar Schuster, who visited Paris-Dauphine last Spring (and is soon to return here as a postdoc funded by Fondation des Sciences Mathématiques de Paris) has arXived last week a paper on gradient importance sampling. In this paper, he builds a sequential importance sampling (or population Monte Carlo) algorithm that exploits the additional information contained in the gradient of the target. The proposal or importance function being essentially the MALA move as its proposal, mixed across the elements of the previous population. When compared with our original PMC mixture of random walk proposals found in e.g. this paper, each term in the mixture thus involves an extra gradient, with a scale factor that decreases to zero as 1/t√t. Ingmar compares his proposal with an adaptive Metropolis, an adaptive MALTa and an HM algorithms, for two mixture distributions and the banana target of Haario et al. (1999) we also used in our paper. As well as a logistic regression. In each case, he finds both a smaller squared error and a smaller bias for the same computing time (evaluated as the number of likelihood evaluations). While we discussed this scheme when he visited, I remain intrigued as to why it works so well when compared with the other solutions. One possible explanation is that the use of the gradient drift is more efficient on a population of particles than on a single Markov chain, provided the population covers all modes of importance on the target surface: the “fatal” attraction of the local model is then much less of an issue…

## Bayesian model averaging in astrophysics

Posted in Books, Statistics, University life with tags , , , , , , , , , , on July 29, 2015 by xi'an

[A 2013 post that somewhat got lost in a pile of postponed entries and referee’s reports…]

In this review paper, now published in Statistical Analysis and Data Mining 6, 3 (2013), David Parkinson and Andrew R. Liddle go over the (Bayesian) model selection and model averaging perspectives. Their argument in favour of model averaging is that model selection via Bayes factors may simply be too inconclusive to favour one model and only one model. While this is a correct perspective, this is about it for the theoretical background provided therein. The authors then move to the computational aspects and the first difficulty is their approximation (6) to the evidence

$P(D|M) = E \approx \frac{1}{n} \sum_{i=1}^n L(\theta_i)Pr(\theta_i)\, ,$

where they average the likelihood x prior terms over simulations from the posterior, which does not provide a valid (either unbiased or converging) approximation. They surprisingly fail to account for the huge statistical literature on evidence and Bayes factor approximation, incl. Chen, Shao and Ibrahim (2000). Which covers earlier developments like bridge sampling (Gelman and Meng, 1998).

As often the case in astrophysics, at least since 2007, the authors’ description of nested sampling drifts away from perceiving it as a regular Monte Carlo technique, with the same convergence speed n1/2 as other Monte Carlo techniques and the same dependence on dimension. It is certainly not the only simulation method where the produced “samples, as well as contributing to the evidence integral, can also be used as posterior samples.” The authors then move to “population Monte Carlo [which] is an adaptive form of importance sampling designed to give a good estimate of the evidence”, a particularly restrictive description of a generic adaptive importance sampling method (Cappé et al., 2004). The approximation of the evidence (9) based on PMC also seems invalid:

$E \approx \frac{1}{n} \sum_{i=1}^n \dfrac{L(\theta_i)}{q(\theta_i)}\, ,$

is missing the prior in the numerator. (The switch from θ in Section 3.1 to X in Section 3.4 is  confusing.) Further, the sentence “PMC gives an unbiased estimator of the evidence in a very small number of such iterations” is misleading in that PMC is unbiased at each iteration. Reversible jump is not described at all (the supposedly higher efficiency of this algorithm is far from guaranteed when facing a small number of models, which is the case here, since the moves between models are governed by a random walk and the acceptance probabilities can be quite low).

The second quite unrelated part of the paper covers published applications in astrophysics. Unrelated because the three different methods exposed in the first part are not compared on the same dataset. Model averaging is obviously based on a computational device that explores the posteriors of the different models under comparison (or, rather, averaging), however no recommendation is found in the paper as to efficiently implement the averaging or anything of the kind. In conclusion, I thus find this review somehow anticlimactic.

## 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.