dynamic mixtures and frequentist ABC

This early morning in NYC, I spotted this new arXival by Marco Bee (whom I know from the time he was writing his PhD with my late friend Bernhard Flury) and found he has been working for a while on ABC related problems. The mixture model he considers therein is a form of mixture of experts, where the weights of the mixture components are not constant but functions on (0,1) of the entry as well. This model was introduced by Frigessi, Haug and Rue in 2002 and is often used as a benchmark for ABC methods, since it is missing its normalising constant as in e.g.

even with all entries being standard pdfs and cdfs. Rather than using a (costly) numerical approximation of the “constant” (as a function of all unknown parameters involved), Marco follows the approximate maximum likelihood approach of my Warwick colleagues, Javier Rubio [now at UCL] and Adam Johansen. It is based on the [SAME] remark that under a uniform prior and using an approximation to the actual likelihood the MAP estimator is also the MLE for that approximation. The approximation is ABC-esque in that a pseudo-sample is generated from the true model (attached to a simulation of the parameter) and the pair is accepted if the pseudo-sample stands close enough to the observed sample. The paper proposes to use the Cramér-von Mises distance, which only involves ranks. Given this “posterior” sample, an approximation of the posterior density is constructed and then numerically optimised. From a frequentist view point, a direct estimate of the mode would be preferable. From my Bayesian perspective, this sounds like a step backwards, given that once a posterior sample is available, reconnecting with an approximate MLE does not sound highly compelling.

truncated mixtures

A question on X validated about EM steps for a truncated Normal mixture led me to ponder whether or not a more ambitious completion [more ambitious than the standard component allocation] was appropriate. Namely, if the mixture is truncated to the interval (a,b), with an observed sample x of size n, this sample could be augmented into an untrucated sample y by latent samples over the complement of (a,b), with random sizes corresponding to the probabilities of falling within (-∞,a), (a,b), and (b,∞). In other words, y is made of three parts, including x, with sizes N¹, n, N³, respectively, the vector (N¹, n, N³) being a trinomial M(N⁺,p) random variable and N⁺ an extra unknown in the model. Assuming a (pseudo-) conjugate prior, an approximate Gibbs sampler can be run (by ignoring the dependence of p on the mixture parameters!). I did not go as far as implementing the idea for the mixture, but had a quick try for a simple truncated Normal. And did not spot any explosive behaviour in N⁺, which is what I was worried about.  Of course, this is mostly anecdotal since the completion does not bring a significant improvement in coding or convergence (the plots corresponds to 10⁴ simulations, for a sample of size n=400).

ABC-SAEM

In connection with the recent PhD thesis defence of Juliette Chevallier, in which I took a somewhat virtual part for being physically in Warwick, I read a paper she wrote with Stéphanie Allassonnière on stochastic approximation versions of the EM algorithm. Computing the MAP estimator can be done via some adapted for simulated annealing versions of EM, possibly using MCMC as for instance in the Monolix software and its MCMC-SAEM algorithm. Where SA stands sometimes for stochastic approximation and sometimes for simulated annealing, originally developed by Gilles Celeux and Jean Diebolt, then reframed by Marc Lavielle and Eric Moulines [friends and coauthors]. With an MCMC step because the simulation of the latent variables involves an untractable normalising constant. (Contrary to this paper, Umberto Picchini and Adeline Samson proposed in 2015 a genuine ABC version of this approach, paper that I thought I missed—although I now remember discussing it with Adeline at JSM in Seattle—, ABC is used as a substitute for the conditional distribution of the latent variables given data and parameter. To be used as a substitute for the Q step of the (SA)EM algorithm. One more approximation step and one more simulation step and we would reach a form of ABC-Gibbs!) In this version, there are very few assumptions made on the approximation sequence, except that it converges with the iteration index to the true distribution (for a fixed observed sample) if convergence of ABC-SAEM is to happen. The paper takes as an illustrative sequence a collection of tempered versions of the true conditionals, but this is quite formal as I cannot fathom a feasible simulation from the tempered version and not from the untempered one. It is thus much more a version of tempered SAEM than truly connected with ABC (although a genuine ABC-EM version could be envisioned).

reaching transcendence for Gaussian mixtures

“…likelihood inference is in a fundamental way more complicated than the classical method of moments.”

Carlos Amendola, Mathias Drton, and Bernd Sturmfels arXived a paper this Friday on “maximum likelihood estimates for Gaussian mixtures are transcendental”. By which they mean that trying to solve the five likelihood equations for a two-component Gaussian mixture does not lead to an algebraic function of the data. (When excluding the trivial global maxima spiking at any observation.) This is not highly surprising when considering two observations, 0 and x, from a mixture of N(0,1/2) and N(μ,1/2) because the likelihood equation

involves both exponential and algebraic terms. While this is not directly impacting (statistical) inference, this result has the computational consequence that the number of critical points ‘and also the maximum number of local maxima, depends on the sample size and increases beyond any bound’, which means that EM faces increasing difficulties in finding a global finite maximum as the sample size increases…

Bangalore workshop [ಬೆಂಗಳೂರು ಕಾರ್ಯಾಗಾರ] and new book

On the last day of the IFCAM workshop in Bangalore, Marc Lavielle from INRIA presented a talk on mixed effects where he illustrated his original computer language Monolix. And mentioned that his CRC Press book on Mixed Effects Models for the Population Approach was out! (Appropriately listed as out on a 14th of July on amazon!) He actually demonstrated the abilities of Monolix live and on diabets data provided by an earlier speaker from Kolkata, which was a perfect way to start initiating a collaboration! Nice cover (which is all I saw from the book at this stage!) that maybe will induce candidates to write a review for CHANCE. Estimation of those mixed effect models relies on stochastic EM algorithms developed by Marc Lavielle and Éric Moulines in the 90’s, as well as MCMC methods.