Archive for non-local priors

ISBA 18 tidbits

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on July 2, 2018 by xi'an

Among a continuous sequence of appealing sessions at this ISBA 2018 meeting [says a member of the scientific committee!], I happened to attend two talks [with a wee bit of overlap] by Sid Chib in two consecutive sessions, because his co-author Ana Simoni (CREST) was unfortunately sick. Their work was about models defined by a collection of moment conditions, as often happens in econometrics, developed in a recent JASA paper by Chib, Shin, and Simoni (2017). With an extension about moving to defining conditional expectations by use of a functional basis. The main approach relies on exponentially tilted empirical likelihoods, which reminded me of the empirical likelihood [BCel] implementation we ran with Kerrie Mengersen and Pierre Pudlo a few years ago. As a substitute to ABC. This problematic made me wonder on how much Bayesian the estimating equation concept is, as it should somewhat involve a nonparametric prior under the moment constraints.

Note that Sid’s [talks and] papers are disconnected from ABC, as everything comes in closed form, apart from the empirical likelihood derivation, as we actually found in our own work!, but this could become a substitute model for ABC uses. For instance, identifying the parameter θ of the model by identifying equations. Would that impose too much input from the modeller? I figure I came with this notion mostly because of the emphasis on proxy models the previous day at ABC in ‘burgh! Another connected item of interest in the work is the possibility of accounting for misspecification of these moment conditions by introducing a vector of errors with a spike & slab distribution, although I am not sure this is 100% necessary without getting further into the paper(s) [blame conference pressure on my time].

Another highlight was attending a fantastic poster session Monday night on computational methods except I would have needed four more hours to get through every and all posters. This new version of ISBA has split the posters between two sites (great) and themes (not so great!), while I would have preferred more sites covering all themes over all nights, to lower the noise (still bearable this year) and to increase the possibility to check all posters of interest in a particular theme…

Mentioning as well a great talk by Dan Roy about assessing deep learning performances by what he calls non-vacuous error bounds. Namely, through PAC-Bayesian bounds. One major comment of his was about deep learning models being much more non-parametric (number of parameters rising with number of observations) than parametric models, meaning that generative adversarial constructs as the one I discussed a few days ago may face a fundamental difficulty as models are taken at face value there.

On closed-form solutions, a closed-form Bayes factor for component selection in mixture models by Fũqene, Steel and Rossell that resemble the Savage-Dickey version, without the measure theoretic difficulties. But with non-local priors. And closed-form conjugate priors for the probit regression model, using unified skew-normal priors, as exhibited by Daniele Durante. Which are product of Normal cdfs and pdfs, and which allow for closed form marginal likelihoods and marginal posteriors as well. (The approach is not exactly conjugate as the prior and the posterior are not in the same family.)

And on the final session I attended, there were two talks on scalable MCMC, one on coresets, which will require some time and effort to assimilate, by Trevor Campbell and Tamara Broderick, and another one using Poisson subsampling. By Matias Quiroz and co-authors. Which did not completely convinced me (but this was the end of a long day…)

All in all, this has been a great edition of the ISBA meetings, if quite intense due to a non-stop schedule, with a very efficient organisation that made parallel sessions manageable and poster sessions back to a reasonable scale [although I did not once manage to cross the street to the other session]. Being in unreasonably sunny Edinburgh helped a lot obviously! I am a wee bit disappointed that no one else follows my call to wear a kilt, but I had low expectations to start with… And too bad I missed the Ironman 70.3 Edinburgh by one day!

non-local priors for mixtures

Posted in Statistics, University life with tags , , , , , , , , , , , , , , , on September 15, 2016 by xi'an

[For some unknown reason, this commentary on the paper by Jairo Fúquene, Mark Steel, David Rossell —all colleagues at Warwick— on choosing mixture components by non-local priors remained untouched in my draft box…]

Choosing the number of components in a mixture of (e.g., Gaussian) distributions is a hard problem. It may actually be an altogether impossible problem, even when abstaining from moral judgements on mixtures. I do realise that the components can eventually be identified as the number of observations grows to infinity, as demonstrated foFaith, Barossa Valley wine: strange name for a Shiraz (as it cannot be a mass wine!, but nice flavoursr instance by Judith Rousseau and Kerrie Mengersen (2011). But for a finite and given number of observations, how much can we trust any conclusion about the number of components?! It seems to me that the criticism about the vacuity of point null hypotheses, namely the logical absurdity of trying to differentiate θ=0 from any other value of θ, applies to the estimation or test on the number of components of a mixture. Doubly so, one might argue, since a very small or a very close component is undistinguishable from a non-existing one. For instance, Definition 2 is correct from a mathematical viewpoint, but it does not spell out the multiple contiguities between k and k’ component mixtures.

The paper starts with a comprehensive coverage of l’état de l’art… When using a Bayes factor to compare a k-component and an h-component mixture, the behaviour of the factor is quite different depending on which model is correct. Essentially overfitted mixtures take much longer to detect than underfitted ones, which makes intuitive sense. And BIC should be corrected for overfitted mixtures by a canonical dimension λ between the true and the (larger) assumed number of parameters  into

2 log m(y) = 2 log p(y|θ) – λ log O(n) + O(log log n)

I would argue that this purely invalidates BIG in mixture settings since the canonical dimension λ is unavailable (and DIC does not provide a useful substitute as we illustrated a decade ago…) The criticism about Rousseau and Mengersen (2011) over-fitted mixture that their approach shrinks less than a model averaging over several numbers of components relates to minimaxity and hence sounds both overly technical and reverting to some frequentist approach to testing. Replacing testing with estimating sounds like the right idea.  And I am also unconvinced that a faster rate of convergence of the posterior probability or of the Bayes factor is a relevant factor when conducting

As for non local priors, the notion seems to rely on a specific topology for the parameter space since a k-component mixture can approach a k’-component mixture (when k'<k) in a continuum of ways (even for a given parameterisation). This topology seems to be summarised by the penalty (distance?) d(θ) in the paper. Is there an intrinsic version of d(θ), given the weird parameter space? Like one derived from the Kullback-Leibler distance between the models? The choice of how zero is approached clearly has an impact on how easily the “null” is detected, the more because of the somewhat discontinuous nature of the parameter space. Incidentally, I find it curious that only the distance between means is penalised… The prior also assumes independence between component parameters and component weights, which I think is suboptimal in dealing with mixtures, maybe suboptimal in a poetic sense!, as we discussed in our reparameterisation paper. I am not sure either than the speed the distance converges to zero (in Theorem 1) helps me to understand whether the mixture has too many components for the data’s own good when I can run a calibration experiment under both assumptions.

While I appreciate the derivation of a closed form non-local prior, I wonder at the importance of the result. Is it because this leads to an easier derivation of the posterior probability? I do not see the connection in Section 3, except maybe that the importance weight indeed involves this normalising constant when considering several k’s in parallel. Is there any convergence issue in the importance sampling solution of (3.1) and (3.3) since the simulations are run under the local posterior? While I appreciate the availability of an EM version for deriving the MAP, a fact I became aware of only recently, is it truly bringing an improvement when compared with picking the MCMC simulation with the highest completed posterior?

The section on prior elicitation is obviously of central interest to me! It however seems to be restricted to the derivation of the scale factor g, in the distance, and of the parameter q in the Dirichlet prior on the weights. While the other parameters suffer from being allocated the conjugate-like priors. I would obviously enjoy seeing how this approach proceeds with our non-informative prior(s). In this regard, the illustration section is nice, but one always wonders at the representative nature of the examples and the possible interpretations of real datasets. For instance, when considering that the Old Faithful is more of an HMM than a mixture.

twilight zone [of statistics]

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , , , on February 26, 2016 by xi'an

mixture with unknown means“I have decided that mixtures, like tequila, are inherently evil and should be avoided at all costs.” L. Wasserman

Larry Wasserman once remarked that finite mixtures were like the twilight zone of statistics, thanks to the numerous idiosyncrasies associated with such models. And George Casella had similar strong reservations about mixture estimation. Avi Feller and co-authors [including Natesh Pillai] have just arXived a paper on this topic, exhibiting shocking (!) properties of the MLE! Their core example is a mixture of two normal distributions with known common variance and known weight different from 0.5, which ensures identifiability. This is a favourite example of mine that we used for instance in our book Introducing Monte Carlo methods with R. If only because we can plot the likelihood and posterior surfaces. (Warning: I wrote those notes on an earlier version of the paper, so mileage may vary in terms of accuracy!)

The “shocking” discovery in the paper is that the MLE is wrong as often as not in selecting the sign of the difference Δ between both means, with an additional accumulation point at zero. The global mode may thus be in the wrong place for small enough sample sizes. And even for larger sizes: when the difference between the means is small the likelihood is likely to be unimodal with a mode quite close to zero. (An interesting remark is that the likelihood derivative is always zero at Δ=0 when considering the special case of both means equal to -Δ and to πΔ/(1-π), respectively, which implies that the overall mean of the mixture is equal to zero. A potential connection with our reparameterisation paper, maybe?)

The alternative proposed by Avi and his co-authors is to proceed through moments, i.e., to revert to Pearson (1892). There are however difficulties with this approach, first and foremost the non-uniqueness of the moment equations used to estimate Δ. For instance, the second cumulant equation chosen by the authors is not always defined as opposed to the third cumulant equation (why not using this third cumulant then). Which does not always produce the right sign… But, in a strange twist, the authors turn those deficiencies into signals for both pathologies (wrong sign and “pile-up” at zero).

“…the grid bootstrap yields an exact p-value for any valid test statistic.”

The most importance issue in this framework being in estimating the parameters, the authors opt for an approach based on tests, which is definitely surprising given the well-known deficiencies of standard tests in mixtures. The test chosen here is a Wald test with a statistic equal to the χ² version of the first cumulant differences. I am surprised that the χ² approximation works in such an unfriendly setting. And I do not understand how the grid is used, unless a certain degree of approximation is accepted, which takes us back to the “dark ages” of imposing a minimal distance Δ to achieve consistency, as in Ghosh and Sen (1985).

muminusmu0 muminusmu1

“..our concern about sign error is trivial in the Bayesian setting: the global mode is simply a poor summary of a multi-modal posterior. More broadly, the weak identification issues we highlight in this paper are not necessarily relevant to a strict Bayesian.”

A priori, I do not think pathologies of the MLE always transfer to Bayes estimators, unless one uses the MAP as an [poor] estimator. But using the MAP is not necessary since posterior means are meaningful in this identified setting, where label switching should not occur. However, running the same experiments with a Gaussian prior on both means and using the posterior mean as my estimator, I did obtain the same pathology of Bayes estimates [also produced in the supplementary material] not concentrating on the true value of the difference, but putting weight on the opposite value and at zero. Using a less standard prior inspired by David Rossell’s talk on non-local priors two weeks ago, which avoids a neighbourhood of zero, I did not get a much different picture as illustrated below:

muminusmux0 muminusmux0

Overall, I remain somewhat uncertain as to what to conclude from this pathological behaviour. When both means are close enough, the sign of the difference is often estimated wrongly. But that could simply mean that the means are not significantly different, for that sample size…

JSM 2015 [day #2]

Posted in Books, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , on August 11, 2015 by xi'an

Today, at JSM 2015, in Seattle, I attended several Bayesian sessions, having sadly missed the Dennis Lindley memorial session yesterday, as it clashed with my own session. In the morning sessions on Bayesian model choice, David Rossell (Warwick) defended non-local priors à la Johnson (& Rossell) as having better frequentist properties. Although I appreciate the concept of eliminating a neighbourhood of the null in the alternative prior, even from a Bayesian viewpoint since it forces us to declare explicitly when the null is no longer acceptable, I find the asymptotic motivation for the prior less commendable and open to arbitrary choices that may lead to huge variations in the numerical value of the Bayes factor. Another talk by Jin Wang merged spike and slab with EM with bootstrap with random forests in variable selection. But I could not fathom what the intended properties of the method were… Besides returning another type of MAP.

The second Bayesian session of the morn was mostly centred on sparsity and penalisation, with Carlos Carvalho and Rob McCulloch discussing a two step method that goes through a standard posterior  construction on the saturated model, before using a utility function to select the pertinent variables. Separation of utility from prior was a novel concept for me, if not for Jay Kadane who objected to Rob a few years ago that he put in the prior what should be in the utility… New for me because I always considered the product prior x utility as the main brick in building the Bayesian edifice… Following Herman Rubin’s motto! Veronika Rocková linked with this post-LASSO perspective by studying spike & slab priors based on Laplace priors. While Veronicka’s goal was to achieve sparsity and consistency, this modelling made me wonder at the potential equivalent in our mixtures for testing approach. I concluded that having a mixture of two priors could be translated in a mixture over the sample with two different parameters, each with a different prior. A different topic, namely multiple testing, was treated by Jim Berger, who showed convincingly in my opinion that a Bayesian approach provides a significant advantage.

In the afternoon finalists of the ISBA Savage Award presented their PhD work, both in the theory and  methods section and in the application section. Besides Veronicka Rocková’s work on a Bayesian approach to factor analysis, with a remarkable resolution via a non-parametric Indian buffet prior and a variable selection interpretation that avoids MCMC difficulties, Vinayak Rao wrote his thesis on MCMC methods for jump processes with a finite number of observations, using a highly convincing completion scheme that created independence between blocks and which reminded me of the Papaspiliopoulos et al. (2005) trick for continuous time processes. I do wonder at the potential impact of this method for processing the coalescent trees in population genetics. Two talks dealt with inference on graphical models, Masanao Yajima and  Christine Peterson, inferring the structure of a sparse graph by Bayesian methods.  With applications in protein networks. And with again a spike & slab prior in Christine’s work. The last talk by Sayantan Banerjee was connected to most others in this Savage session in that it also dealt with sparsity. When estimating a large covariance matrix. (It is always interesting to try to spot tendencies in awards and conferences. Following the Bayesian non-parametric era, are we now entering the Bayesian sparsity era? We will see if this is the case at ISBA 2016!) And the winner is..?! We will know tomorrow night! In the meanwhile, congrats to my friends Sudipto Banerjee, Igor Prünster, Sylvia Richardson, and Judith Rousseau who got nominated IMS Fellows tonight.

Cancún, ISBA 2014 [day #3]

Posted in pictures, Statistics, Travel, University life with tags , , , , , on July 23, 2014 by xi'an

Cancun13…already Thursday, our [early] departure day!, with an nth (!) non-parametric session that saw [the newly elected ISBA Fellow!] Judith Rousseau present an ongoing work with Chris Holmes on the convergence or non-convergence conditions for a Bayes factor of a non-parametric hypothesis against another non-parametric. I wondered at the applicability of this test as the selection criterion in ABC settings, even though having an iid sample to start with is a rather strong requirement.

Switching between a scalable computation session with Alex Beskos, who talked about adaptive Langevin algorithms for differential equations, and a non-local prior session, with David Rossell presenting a smoother way to handle point masses in order to accommodate frequentist coverage. Something we definitely need to discuss the next time I am in Warwick! Although this made me alas miss both the first talk of the non-local session by Shane Jensen  the final talk of the scalable session by Doug Vandewrken where I happened to be quoted (!) for my warning about discretising Markov chains into non-Markov processes. In the 1998 JASA paper with Chantal Guihenneuc.

After a farewell meal of ceviche with friends in the sweltering humidity of a local restaurant, I attended [the newly elected ISBA Fellow!] Maria Vanucci’s talk on her deeply involved modelling of fMRI. The last talk before the airport shuttle was François Caron’s description of a joint work with Emily Fox on a sparser modelling of networks, along with an auxiliary variable approach that allowed for parallelisation of a Gibbs sampler. François mentioned an earlier alternative found in machine learning where all components of a vector are updated simultaneously conditional on the previous avatar of the other components, e.g. simulating (x’,y’) from π(x’|y) π(y’|x) which does not produce a convergent Markov chain. At least not convergent to the right stationary. However, running a quick [in-flight] check on a 2-d normal target did not show any divergent feature, when compared with the regular Gibbs sampler. I thus wonder at what can be said about the resulting target or which conditions are need for divergence. A few scribbles later, I realised that the 2-d case was the exception, namely that the stationary distribution of the chain is the product of the marginal. However, running a 3-d example with an auto-exponential distribution in the taxi back home, I still could not spot a difference in the outcome.