Archive for Dirichlet process

slice sampling for Dirichlet mixture process

Posted in Books, Statistics, University life with tags , , , , , , , on June 21, 2017 by xi'an

When working with my PhD student Changye in Dauphine this morning I realised that slice sampling also applies to discrete support distributions and could even be of use in such settings. That it works is (now) straightforward in that the missing variable representation behind the slice sampler also applies to densities defined with respect to a discrete measure. That this is useful transpires from the short paper of Stephen Walker (2007) where we saw this, as Stephen relies on the slice sampler to sample from the Dirichlet mixture model by eliminating the tail problem associated with this distribution. (This paper appeared in Communications in Statistics and it is through Pati & Dunson (2014) taking advantage of this trick that Changye found about its very existence. I may have known about it in an earlier life, but I had clearly forgotten everything!)

While the prior distribution (of the weights) of the Dirichlet mixture process is easy to generate via the stick breaking representation, the posterior distribution is trickier as the weights are multiplied by the values of the sampling distribution (likelihood) at the corresponding parameter values and they cannot be normalised. Introducing a uniform to replace all weights in the mixture with an indicator that the uniform is less than those weights corresponds to a (latent variable) completion [or a demarginalisation as we called this trick in Monte Carlo Statistical Methods]. As elaborated in the paper, the Gibbs steps corresponding to this completion are easy to implement, involving only a finite number of components. Meaning the allocation to a component of the mixture can be operated rather efficiently. Or not when considering that the weights in the Dirichlet mixture are not monotone, hence that a large number of them may need to be computed before picking the next index in the mixture when the uniform draw happens to be quite small.

comments on Watson and Holmes

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , , on April 1, 2016 by xi'an


“The world is full of obvious things which nobody by any chance ever observes.” The Hound of the Baskervilles

In connection with the incoming publication of James Watson’s and Chris Holmes’ Approximating models and robust decisions in Statistical Science, Judith Rousseau and I wrote a discussion on the paper that has been arXived yesterday.

“Overall, we consider that the calibration of the Kullback-Leibler divergence remains an open problem.” (p.18)

While the paper connects with earlier ones by Chris and coauthors, and possibly despite the overall critical tone of the comments!, I really appreciate the renewed interest in robustness advocated in this paper. I was going to write Bayesian robustness but to differ from the perspective adopted in the 90’s where robustness was mostly about the prior, I would say this is rather a Bayesian approach to model robustness from a decisional perspective. With definitive innovations like considering the impact of posterior uncertainty over the decision space, uncertainty being defined e.g. in terms of Kullback-Leibler neighbourhoods. Or with a Dirichlet process distribution on the posterior. This may step out of the standard Bayesian approach but it remains of definite interest! (And note that this discussion of ours [reluctantly!] refrained from capitalising on the names of the authors to build easy puns linked with the most Bayesian of all detectives!)

Dirichlet process mixture inconsistency

Posted in Books, Statistics with tags , , , , on February 15, 2016 by xi'an

cover of Mixture Estimation and ApplicationsJudith Rousseau pointed out to me this NIPS paper by Jeff Miller and Matthew Harrison on the possible inconsistency of Dirichlet mixtures priors for estimating the (true) number of components in a (true) mixture model. The resulting posterior on the number of components does not concentrate on the right number of components. Which is not the case when setting a prior on the unknown number of components of a mixture, where consistency occurs. (The inconsistency results established in the paper are actually focussed on iid Gaussian observations, for which the estimated number of Gaussian components is almost never equal to 1.) In a more recent arXiv paper, they also show that a Dirichlet prior on the weights and a prior on the number of components can still produce the same features as a Dirichlet mixtures priors. Even the stick breaking representation! (Paper that I already reviewed last Spring.)

Conditional love [guest post]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , on August 4, 2015 by xi'an

[When Dan Simpson told me he was reading Terenin’s and Draper’s latest arXival in a nice Bath pub—and not a nice bath tub!—, I asked him for a blog entry and he agreed. Here is his piece, read at your own risk! If you remember to skip the part about Céline Dion, you should enjoy it very much!!!]

Probability has traditionally been described, as per Kolmogorov and his ardent follower Katy Perry, unconditionally. This is, of course, excellent for those of us who really like measure theory, as the maths is identical. Unfortunately mathematical convenience is not necessarily enough and a large part of the applied statistical community is working with Bayesian methods. These are unavoidably conditional and, as such, it is natural to ask if there is a fundamentally conditional basis for probability.

Bruno de Finetti—and later Richard Cox and Edwin Jaynes—considered conditional bases for Bayesian probability that are, unfortunately, incomplete. The critical problem is that they mainly consider finite state spaces and construct finitely additive systems of conditional probability. For a variety of reasons, neither of these restrictions hold much truck in the modern world of statistics.

In a recently arXiv’d paper, Alexander Terenin and David Draper devise a set of axioms that make the Cox-Jaynes system of conditional probability rigorous. Furthermore, they show that the complete set of Kolmogorov axioms (including countable additivity) can be derived as theorems from their axioms by conditioning on the entire sample space.

This is a deep and fundamental paper, which unfortunately means that I most probably do not grasp it’s complexities (especially as, for some reason, I keep reading it in pubs!). However I’m going to have a shot at having some thoughts on it, because I feel like it’s the sort of paper one should have thoughts on. Continue reading

mixture models with a prior on the number of components

Posted in Books, Statistics, University life with tags , , , , , , , on March 6, 2015 by xi'an

mixdir

“From a Bayesian perspective, perhaps the most natural approach is to treat the numberof components like any other unknown parameter and put a prior on it.”

Another mixture paper on arXiv! Indeed, Jeffrey Miller and Matthew Harrison recently arXived a paper on estimating the number of components in a mixture model, comparing the parametric with the non-parametric Dirichlet prior approaches. Since priors can be chosen towards agreement between those. This is an obviously interesting issue, as they are often opposed in modelling debates. The above graph shows a crystal clear agreement between finite component mixture modelling and Dirichlet process modelling. The same happens for classification.  However, Dirichlet process priors do not return an estimate of the number of components, which may be considered a drawback if one considers this is an identifiable quantity in a mixture model… But the paper stresses that the number of estimated clusters under the Dirichlet process modelling tends to be larger than the number of components in the finite case. Hence that the Dirichlet process mixture modelling is not consistent in that respect, producing parasite extra clusters…

In the parametric modelling, the authors assume the same scale is used in all Dirichlet priors, that is, for all values of k, the number of components. Which means an incoherence when marginalising from k to (k-p) components. Mild incoherence, in fact, as the parameters of the different models do not have to share the same priors. And, as shown by Proposition 3.3 in the paper, this does not prevent coherence in the marginal distribution of the latent variables. The authors also draw a comparison between the distribution of the partition in the finite mixture case and the Chinese restaurant process associated with the partition in the infinite case. A further analogy is that the finite case allows for a stick breaking representation. A noteworthy difference between both modellings is about the size of the partitions

\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-\gamma}\quad\text{versus}\quad\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-1}

in the finite (homogeneous partitions) and infinite (extreme partitions) cases.

An interesting entry into the connections between “regular” mixture modelling and Dirichlet mixture models. Maybe not ultimately surprising given the past studies by Peter Green and Sylvia Richardson of both approaches (1997 in Series B and 2001 in JASA).

improved approximate-Bayesian model-choice method for estimating shared evolutionary history [reply from the author]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on June 3, 2014 by xi'an

[Here is a very kind and detailed reply from Jamie Oakes to the comments I made on his ABC paper a few days ago:]

First of all, many thanks for your thorough review of my pre-print! It is very helpful and much appreciated. I just wanted to comment on a few things you address in your post.

I am a little confused about how my replacement of continuous uniform probability distributions with gamma distributions for priors on several parameters introduces a potentially crippling number of hyperparameters. Both uniform and gamma distributions have two parameters. So, the new model only has one additional hyperparameter compared to the original msBayes model: the concentration parameter on the Dirichlet process prior on divergence models. Also, the new model offers a uniform prior over divergence models (though I don’t recommend it).

Your comment about there being no new ABC technique is 100% correct. The model is new, the ABC numerical machinery is not. Also, your intuition is correct, I do not use the divergence times to calculate summary statistics. I mention the divergence times in the description of the ABC algorithm with the hope of making it clear that the times are scaled (see Equation (12)) prior to the simulation of the data (from which the summary statistics are calculated). This scaling is simply to go from units proportional to time, to units that are proportional to the expected number of mutations. Clearly, my attempt at clarity only created unnecessary opacity. I’ll have to make some edits.

Regarding the reshuffling of the summary statistics calculated from different alignments of sequences, the statistics are not exchangeable. So, reshuffling them in a manner that is not conistent across all simulations and the observed data is not mathematically valid. Also, if elements are exchangeable, their order will not affect the likelihood (or the posterior, barring sampling error). Thus, if our goal is to approximate the likelihood, I would hope the reshuffling would also have little affect on the approximate posterior (otherwise my approximation is not so good?).

You are correct that my use of “bias” was not well defined in reference to the identity line of my plots of the estimated vs true probability of the one-divergence model. I think we can agree that, ideally (all assumptions are met), the estimated posterior probability of a model should estimate the probability that the model is correct. For large numbers of simulation
replicates, the proportion of the replicates for which the one-divergence model is true will approximate the probability that the one-divergence model is correct. Thus, if the method has the desirable (albeit “frequentist”) behavior such that the estimated posterior probability of the one-divergence model is an unbiased estimate of the probability that the one-divergence model is correct, the points should fall near the identity line. For example, let us say the method estimates a posterior probability of 0.90 for the one-divergence model for 1000 simulated datasets. If the method is accurately estimating the probability that the one-divergence model is the correct model, then the one-divergence model should be the true model for approximately 900 of the 1000 datasets. Any trend away from the identity line indicates the method is biased in the (frequentist) sense that it is not correctly estimating the probability that the one-divergence model is the correct model. I agree this measure of “bias” is frequentist in nature. However, it seems like a worthwhile goal for Bayesian model-choice methods to have good frequentist properties. If a method strongly deviates from the identity line, it is much more difficult to interpret the posterior probabilites that it estimates. Going back to my example of the posterior probability of 0.90 for 1000 replicates, I would be alarmed if the model was true in only 100 of the replicates.

My apologies if my citation of your PNAS paper seemed misleading. The citation was intended to be limited to the context of ABC methods that use summary statistics that are insufficient across the models under comparison (like msBayes and the method I present in the paper). I will definitely expand on this sentence to make this clearer in revisions. Thanks!

Lastly, my concluding remarks in the paper about full-likelihood methods in this domain are not as lofty as you might think. The likelihood function of the msBayes model is tractable, and, in fact, has already been derived and implemented via reversible-jump MCMC (albeit, not readily available yet). Also, there are plenty of examples of rich, Kingman-coalescent models implemented in full-likelihood Bayesian frameworks. Too many to list, but a lot of them are implemented in the BEAST software package. One noteworthy example is the work of Bryant et al. (2012, Molecular Biology and Evolution, 29(8), 1917–32) that analytically integrates over all gene trees for biallelic markers under the coalescent.

improved approximate-Bayesian model-choice method for estimating shared evolutionary history

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on May 14, 2014 by xi'an

“An appealing approach would be a comparative, Bayesian model-choice method for inferring the probability of competing divergence histories while integrating over uncertainty in mutational and ancestral processes via models of nucleotide substitution and lineage coalescence.” (p.2)

Jamies Oaks arXived (a few months ago now) a rather extensive Monte-Carlo study on the impact of prior modelling on the model-choice performances of ABC model choice. (Of which I only became aware recently.) As in the earlier paper I commented on the Óg, the issue here has much more to do with prior assessment and calibration than with ABC implementation per se. For instance, the above quote recaps the whole point of conducting Bayesian model choice. (As missed by Templeton.)

“This causes divergence models with more divergence-time parameters to integrate over a much greater parameter space with low likelihood yet high prior density, resulting in small marginal likelihoods relative to models with fewer divergence-time parameters.” (p.2)

This second quote is essentially stressing the point with Occam’s razor argument. Which I deem [to be] a rather positive feature of Bayesian model choice. A reflection on the determination of the prior distribution, getting away from uniform priors, thus sounds most timely! The current paper takes place within a rather extensive exchange between Oak’s group and Hickerson’s group on what makes Bayesian model choice (and the associated software msBayes) pick or not the correct model. Oak and coauthors objected to the use of “narrow, empirically informed uniform priors”, arguing that this leads to a bias towards models with less parameters, a “statistical issue” in their words, while Hickerson et al. (2014) think this is due to msBayes way of selecting models and their parameters at random. However it refrains from reproducing earlier criticisms of or replies to Hickerson et al.

The current paper claims to have reached a satisfactory prior modelling with ¨improved robustness, accuracy, and power” (p.3).  If I understand correctly, the changes are in replacing a uniform distribution with a Gamma or a Dirichlet prior. Which means introducing a seriously large and potentially crippling number of hyperparameters into the picture. Having a lot of flexibility in the prior also means a lot of variability in the resulting inference… In other words, with more flexibility comes more responsibility, to paraphrase Voltaire.

“I have introduced a new approximate-Bayesian model choice method.” (p.21)

The ABC part is rather standard, except for the strange feature that the divergence times are used to construct summary statistics (p.10). Strange because these times are not observed for the actual data. So I must be missing something. (And I object to the above quote and to the title of the paper since there is no new ABC technique there, simply a different form of prior.)

“ABC methods in general are known to be biased for model choice.” (p.21)

I do not understand much the part about (reshuffling) introducing bias as detailed on p.11: every approximate method gives a “biased” answer in the sense this answer is not the true and proper posterior distribution. Using a different (re-ordered) vector of statistics provides a different ABC outcome,  hence a different approximate posterior, for which it seems truly impossible to check whether or not it increases the discrepancy from the true posterior, compared with the other version. I must admit I always find annoying to see the word bias used in a vague meaning and esp. within a Bayesian setting. All Bayesian methods are biased. End of the story. Quoting our PNAS paper as concluding that ABC model choice is biased is equally misleading: the intended warning represented by the paper was that Bayes factors and posterior probabilities could be quite unrelated with those based on the whole dataset. That the proper choice of summary statistics leads to a consistent model choice shows ABC model choice is not necessarily “biased”… Furthermore, I also fail to understand why the posterior probability of model i should be distributed as a uniform (“If the method is unbiased, the points should fall near the identity line”) when the data is from model i: this is not a p-value but a posterior probability and the posterior probability is not the frequentist coverage…

My overall problem is that, all in all, this is a single if elaborate Monte Carlo study and, as such, it does not carry enough weight to validate an approach that remains highly subjective in the selection of its hyperparameters. Without raising any doubt about an hypothetical “fixing” of those hyperparameters, I think this remains a controlled experiment with simulated data where the true parameters are know and the prior is “true”. This obviously helps in getting better performances.

“With improving numerical methods (…), advances in Monte Carlo techniques and increasing efficiency of likelihood calculations, analyzing rich comparative phylo-geographical models in a full-likelihood Bayesian framework is becoming computationally feasible.” (p.21)

This conclusion of the paper sounds over-optimistic and rather premature. I do not know of any significant advance in computing the observed likelihood for the population genetics models ABC is currently handling. (The SMC algorithm of Bouchard-Côté, Sankaraman and Jordan, 2012, does not apply to Kingman’s coalescent, as far as I can tell.) This is certainly a goal worth pursuing and borrowing strength from multiple techniques cannot hurt, but it remains so far a lofty goal, still beyond our reach… I thus think the major message of the paper is to reinforce our own and earlier calls for caution when interpreting the output of an ABC model choice (p.20), or even of a regular Bayesian analysis, agreeing that we should aim at seeing “a large amount of posterior uncertainty” rather than posterior probability values close to 0 and 1.