## relabelling in Bayesian mixtures by pivotal units

Posted in Statistics with tags , , , , on September 14, 2017 by xi'an

Yet another paper on relabelling for mixtures, when one would think everything and more has already be said and written on the topic… This one appeared in Statistics and Computing last August and I only became aware of it through ResearchGate which sent me an unsolicited email that this paper quoted one of my own papers. As well as Bayesian Essentials.

The current paper by Egidi, Pappadà, Pauli and Torelli starts from the remark that the similarity matrix of the probabilities for pairs of observations to be in the same component is invariant to label switching. A property we also used in our 2000 JASA paper. But here the authors assume it is possible to find pivots, that is, as many observations as there are components such that any pair of them is never in the same component with posterior probability one. These pivots are then used for the relabelling, as they define a preferential relabelling at each iteration. Now, this is not always possible since there are presumably iterations with empty components and there is rarely a zero probability that enough pairs never meet. The resolution of this quandary is then to remove the iterations for which this happens, a subsampling that changes the nature of the MCMC chain and may jeopardise its Markovian validation. The authors however suggest using alternative and computationally cheaper solutions to identify the pivots. (Which confuses me as to which solution they adopt.)

The next part of the paper compares this approach with seven other solutions found in the literature, from Matthew Stephens’ (2000) to our permutation reordering. Which does pretty well in terms of MSE in the simulation study (see the massive Table 3) while being much cheaper to implement than the proposed pivotal relabelling (Table 4). And which, contrary to the authors’ objection, does not require the precise computation of the MAP since, as indicated in our paper, the relative maximum based on the MCMC iterations can be used as a proxy. I am thus less than convinced at the improvement brought by this alternative…

## label switching in Bayesian mixture models

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on October 31, 2014 by xi'an

A referee of our paper on approximating evidence for mixture model with Jeong Eun Lee pointed out the recent paper by Carlos Rodríguez and Stephen Walker on label switching in Bayesian mixture models: deterministic relabelling strategies. Which appeared this year in JCGS and went beyond, below or above my radar.

Label switching is an issue with mixture estimation (and other latent variable models) because mixture models are ill-posed models where part of the parameter is not identifiable. Indeed, the density of a mixture being a sum of terms

$\sum_{j=1}^k \omega_j f(y|\theta_i)$

the parameter (vector) of the ω’s and of the θ’s is at best identifiable up to an arbitrary permutation of the components of the above sum. In other words, “component #1 of the mixture” is not a meaningful concept. And hence cannot be estimated.

This problem has been known for quite a while, much prior to EM and MCMC algorithms for mixtures, but it is only since mixtures have become truly estimable by Bayesian approaches that the debate has grown on this issue. In the very early days, Jean Diebolt and I proposed ordering the components in a unique way to give them a meaning. For instant, “component #1” would then be the component with the smallest mean or the smallest weight and so on… Later, in one of my favourite X papers, with Gilles Celeux and Merrilee Hurn, we exposed the convergence issues related with the non-identifiability of mixture models, namely that the posterior distributions were almost always multimodal, with a multiple of k! symmetric modes in the case of exchangeable priors, and therefore that Markov chains would have trouble to visit all those modes in a symmetric manner, despite the symmetry being guaranteed from the shape of the posterior. And we conclude with the slightly provocative statement that hardly any Markov chain inferring about mixture models had ever converged! In parallel, time-wise, Matthew Stephens had completed a thesis at Oxford on the same topic and proposed solutions for relabelling MCMC simulations in order to identify a single mode and hence produce meaningful estimators. Giving another meaning to the notion of “component #1”.

And then the topic began to attract more and more researchers, being both simple to describe and frustrating in its lack of definitive answer, both from simulation and inference perspectives. Rodriguez’s and Walker’s paper provides a survey on the label switching strategies in the Bayesian processing of mixtures, but its innovative part is in deriving a relabelling strategy. Which consists of finding the optimal permutation (at each iteration of the Markov chain) by minimising a loss function inspired from k-means clustering. Which is connected with both Stephens’ and our [JASA, 2000] loss functions. The performances of this new version are shown to be roughly comparable with those of other relabelling strategies, in the case of Gaussian mixtures. (Making me wonder if the choice of the loss function is not favourable to Gaussian mixtures.) And somehow faster than Stephens’ Kullback-Leibler loss approach.

“Hence, in an MCMC algorithm, the indices of the parameters can permute multiple times between iterations. As a result, we cannot identify the hidden groups that make [all] ergodic averages to estimate characteristics of the components useless.”

One section of the paper puzzles me, albeit it does not impact the methodology and the conclusions. In Section 2.1 (p.27), the authors consider the quantity

$p(z_i=j|{\mathbf y})$

which is the marginal probability of allocating observation i to cluster or component j. Under an exchangeable prior, this quantity is uniformly equal to 1/k for all observations i and all components j, by virtue of the invariance under permutation of the indices… So at best this can serve as a control variate. Later in Section 2.2 (p.28), the above sentence does signal a problem with those averages but it seem to attribute it to MCMC behaviour rather than to the invariance of the posterior (or to the non-identifiability of the components per se). At last, the paper mentions that “given the allocations, the likelihood is invariant under permutations of the parameters and the allocations” (p.28), which is not correct, since eqn. (8)

$f(y_i|\theta_{\sigma(z_i)}) =f(y_i|\theta_{\tau(z_i)})$

does not hold when the two permutations σ and τ give different images of zi

## AMOR at 5000ft in a water tank…

Posted in Mountains, pictures, Statistics, University life with tags , , , , , , , , , , , , , , on November 22, 2012 by xi'an

On Monday, I attended the thesis defence of Rémi Bardenet in Orsay as a member (referee) of his thesis committee. While this was a thesis in computer science, which took place in the Linear Accelerator Lab in Orsay, it was clearly rooted in computational statistics, hence justifying my presence in the committee. The justification (!) for the splashy headline of this post is that Rémi’s work was motivated by the Pierre-Auger experiment on ultra-high-energy cosmic rays, where particles are detected through a network of 1600 water tanks spread over the Argentinian Pampa Amarilla on an area the size of Rhode Island (where I am incidentally going next week).

The part of Rémi’s thesis presented during the defence concentrated on his AMOR algorithm, arXived in a paper written with Olivier Cappé and Gersende Fort. AMOR stands for adaptive Metropolis online relabelling and combines adaptive MCMC techniques with relabelling strategies to fight label-switching (e.g., in mixtures). I have been interested in mixtures for eons (starting in 1987 in Ottawa with applying Titterington, Smith, and Makov to chest radiographs) and in label switching for ages (starting at the COMPSTAT conférence in Bristol in 1998). Rémi’s approach to the label switching problem follows the relabelling path, namely a projection of the original parameter space into a smaller subspace (that is also a quotient space) to avoid permutation invariance and lack of identifiability. (In the survey I wrote with Kate Lee, Jean-Michel Marin and Kerrie Mengersen, we suggest using the mode as a pivot to determine which permutation to use on the components of the mixture.) The paper suggests using an Euclidean distance to a mean determined adaptively, μt, with a quadratic form Σt also determined on-the-go, minimising (Pθ-μt)TΣt(Pθ-μt) over all permutations P at each step of the algorithm. The intuition behind the method is that the posterior over the restricted space should look like a roughly elliptically symmetric distribution, or at least like a unimodal distribution, rather than borrowing bits and pieces from different modes. While I appreciate the technical tour de force represented by the proof of convergence of the AMOR algorithm, I remain somehow sceptical about the approach and voiced the following objections during the defence: first, the assumption that the posterior becomes unimodal under an appropriate restriction is not necessarily realistic. Secondary modes often pop in with real data (as in the counter-example we used in our paper with Alessandra Iacobucci and Jean-Michel Marin). Next, the whole apparatus of fighting multiple modes and non-identifiability, i.e. fighting label switching, is to fall back on posterior means as Bayes estimators. As stressed in our JASA paper with Gilles Celeux and Merrilee Hurn, there is no reason for doing so and there are several reasons for not doing so:

• it breaks down under model specification, i.e., when the number of components is not correct
• it does not improve the speed of convergence but, on the opposite, restricts the space visited by the Markov chain
• it may fall victim to the fatal attraction of secondary modes by fitting too small an ellipse around one of those modes
• it ultimately depends on the parameterisation of the model
• there is no reason for using posterior means in mixture problems, posterior modes or cluster centres can be used instead

I am therefore very much more in favour of producing a posterior distribution that is as label switching as possible (since the true posterior is completely symmetric in this respect). Post-processing the resulting sample can be done by using off-the-shelf clustering in the component space, derived from the point process representation used by Matthew Stephens in his thesis and subsequent papers. It also allows for a direct estimation of the number of components.

In any case, this was a defence worth-attending that led me to think afresh about the label switching problem, with directions worth exploring next month while Kate Lee is visiting from Auckland. Rémi Bardenet is now headed for a postdoc in Oxford, a perfect location to discuss further label switching and to engage into new computational statistics research!