## estimating the marginal likelihood (or an information criterion)

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on December 28, 2019 by xi'an Tory Imai (from Kyoto University) arXived a paper last summer on what first looked like a novel approximation of the marginal likelihood. Based on the variance of thermodynamic integration. The starting argument is that there exists a power 0<t⁰<1 such that the expectation of the logarithm of the product of the prior by the likelihood to the power t⁰ or t⁰-powered likelihood  is equal to the standard log-marginal $\log m(x) = \mathbb{E}^{t^0}[ \log f(X|\theta) ]$

when the expectation is under the posterior corresponding to the t⁰-powered likelihood (rather than the full likelihood). By an application of the mean value theorem. Watanabe’s (2013) WBIC replaces the optimum t⁰ with 1/log(n), n being the sample size. The issue in terms of computational statistics is of course that the error of WBIC (against the true log m(x)) is only characterised as an order of n.

The second part of the paper is rather obscure to me, as the motivation for the real log canonical threshold is missing, even though the quantity is connected with the power likelihood. And the DIC effective dimension. It then goes on to propose a new approximation of sBIC, where s stands for singular, of Drton and Plummer (2017) which I had missed (and may ask my colleague Martin later today at Warwick!). Quickly reading through the later however brings explanations about the real log canonical threshold being simply the effective dimension in Schwarwz’s BIC approximation to the log marginal, $\log m(x) \approx= \log f(x|\hat{\theta}_n) - \lambda \log n +(m-1)\log\log n$

(as derived by Watanabe), where m is called the multiplicity of the real log canonical threshold. Both λ and m being unknown, Drton and Plummer (2017) estimate the above approximation in a Bayesian fashion, which leads to a double indexed marginal approximation for a collection of models. Since this thread leads me further and further from a numerical resolution of the marginal estimation, but brings in a different perspective on mixture Bayesian estimation, I will return to this highly  in a later post. The paper of Imai discusses a different numerical approximation to sBIC, With a potential improvement in computing sBIC. (The paper was proposed as a poster to BayesComp 2020, so I am looking forward discussing it with the author.)

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on April 30, 2019 by xi'an Ziheng Yang and Tianqui Zhu published a paper in PNAS last year that criticises Bayesian posterior probabilities used in the comparison of models under misspecification as “overconfident”. The paper is written from a phylogeneticist point of view, rather than from a statistician’s perspective, as shown by the Editor in charge of the paper [although I thought that, after Steve Fienberg‘s intervention!, a statistician had to be involved in a submission relying on statistics!] a paper , but the analysis is rather problematic, at least seen through my own lenses… With no statistical novelty, apart from looking at the distribution of posterior probabilities in toy examples. The starting argument is that Bayesian model comparison is often reporting posterior probabilities in favour of a particular model that are close or even equal to 1.

“The Bayesian method is widely used to estimate species phylogenies using molecular sequence data. While it has long been noted to produce spuriously high posterior probabilities for trees or clades, the precise reasons for this over confidence are unknown. Here we characterize the behavior of Bayesian model selection when the compared models are misspecified and demonstrate that when the models are nearly equally wrong, the method exhibits unpleasant polarized behaviors,supporting one model with high confidence while rejecting others. This provides an explanation for the empirical observation of spuriously high posterior probabilities in molecular phylogenetics.”

The paper focus on the behaviour of posterior probabilities to strongly support a model against others when the sample size is large enough, “even when” all models are wrong, the argument being apparently that the correct output should be one of equal probability between models, or maybe a uniform distribution of these model probabilities over the probability simplex. Why should it be so?! The construction of the posterior probabilities is based on a meta-model that assumes the generating model to be part of a list of mutually exclusive models. It does not account for cases where “all models are wrong” or cases where “all models are right”. The reported probability is furthermore epistemic, in that it is relative to the measure defined by the prior modelling, not to a promise of a frequentist stabilisation in a ill-defined asymptotia. By which I mean that a 99.3% probability of model M¹ being “true”does not have a universal and objective meaning. (Moderation note: the high polarisation of posterior probabilities was instrumental in our investigation of model choice with ABC tools and in proposing instead error rates in ABC random forests.) The notion that two models are equally wrong because they are both exactly at the same Kullback-Leibler distance from the generating process (when optimised over the parameter) is such a formal [or cartoonesque] notion that it does not make much sense. There is always one model that is slightly closer and eventually takes over. It is also bizarre that the argument does not account for the complexity of each model and the resulting (Occam’s razor) penalty. Even two models with a single parameter are not necessarily of intrinsic dimension one, as shown by DIC. And thus it is not a surprise if the posterior probability mostly favours one versus the other. In any case, an healthily sceptic approach to Bayesian model choice means looking at the behaviour of the procedure (Bayes factor, posterior probability, posterior predictive, mixture weight, &tc.) under various assumptions (model M¹, M², &tc.) to calibrate the numerical value, rather than taking it at face value. By which I do not mean a frequentist evaluation of this procedure. Actually, it is rather surprising that the authors of the PNAS paper do not jump on the case when the posterior probability of model M¹ say is uniformly distributed, since this would be a perfect setting when the posterior probability is a p-value. (This is also what happens to the bootstrapped version, see the last paragraph of the paper on p.1859, the year Darwin published his Origin of Species.)

## 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 fo r 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.

## model selection and multiple testing

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on October 23, 2015 by xi'an Ritabrata Dutta, Malgorzata Bogdan and Jayanta Ghosh recently arXived a survey paper on model selection and multiple testing. Which provides a good opportunity to reflect upon traditional Bayesian approaches to model choice. And potential alternatives. On my way back from Madrid, where I got a bit distracted when flying over the South-West French coast, from Biarritz to Bordeaux. Spotting the lake of Hourtain, where I spent my military training month, 29 years ago!

“On the basis of comparison of AIC and BIC, we suggest tentatively that model selection rules should be used for the purpose for which they were introduced. If they are used for other problems, a fresh justification is desirable. In one case, justification may take the form of a consistency theorem, in the other some sort of oracle inequality. Both may be hard to prove. Then one should have substantial numerical assessment over many different examples.”

The authors quickly replace the Bayes factor with BIC, because it is typically consistent. In the comparison between AIC and BIC they mention the connundrum of defining a prior on a nested model from the prior on the nesting model, a problem that has not been properly solved in my opinion. The above quote with its call to a large simulation study reminded me of the paper by Arnold & Loeppky about running such studies through ecdfs. That I did not see as solving the issue. The authors also discuss DIC and Lasso, without making much of a connection between those, or with the above. And then reach the parametric empirical Bayes approach to model selection exemplified by Ed George’s and Don Foster’s 2000 paper. Which achieves asymptotic optimality for posterior prediction loss (p.9). And which unifies a wide range of model selection approaches.

A second part of the survey considers the large p setting, where BIC is not a good approximation to the Bayes factor (when testing whether or not all mean entries are zero). And recalls that there are priors ensuring consistency for the Bayes factor in this very [restrictive] case. Then, in Section 4, the authors move to what they call “cross-validatory Bayes factors”, also known as partial Bayes factors and pseudo-Bayes factors, where the data is split to (a) make the improper prior proper and (b) run the comparison or test on the remaining data. They also show the surprising result that, provided the fraction of the data used to proper-ise the prior does not converge to one, the X validated Bayes factor remains consistent [for the special case above]. The last part of the paper concentrates on multiple testing but is more tentative and conjecturing about convergence results, centring on the differences between full Bayes and empirical Bayes. Then the plane landed in Paris and I stopped my reading, not feeling differently about the topic than when the plane started from Madrid.

## a unified treatment of predictive model comparison

Posted in Books, Statistics, University life with tags , , , , , , , , , on June 16, 2015 by xi'an

“Applying various approximation strategies to the relative predictive performance derived from predictive distributions in frequentist and Bayesian inference yields many of the model comparison techniques ubiquitous in practice, from predictive log loss cross validation to the Bayesian evidence and Bayesian information criteria.”

Michael Betancourt (Warwick) just arXived a paper formalising predictive model comparison in an almost Bourbakian sense! Meaning that he adopts therein a very general representation of the issue, with minimal assumptions on the data generating process (excluding a specific metric and obviously the choice of a testing statistic). He opts for an M-open perspective, meaning that this generating process stands outside the hypothetical statistical model or, in Lindley’s terms, a small world. Within this paradigm, the only way to assess the fit of a model seems to be through the predictive performances of that model. Using for instance an f-divergence like the Kullback-Leibler divergence, based on the true generated process as the reference. I think this however puts a restriction on the choice of small worlds as the probability measure on that small world has to be absolutely continuous wrt the true data generating process for the distance to be finite. While there are arguments in favour of absolutely continuous small worlds, this assumes a knowledge about the true process that we simply cannot gather. Ignoring this difficulty, a relative Kullback-Leibler divergence can be defined in terms of an almost arbitrary reference measure. But as it still relies on the true measure, its evaluation proceeds via cross-validation “tricks” like jackknife and bootstrap. However, on the Bayesian side, using the prior predictive links the Kullback-Leibler divergence with the marginal likelihood. And Michael argues further that the posterior predictive can be seen as the unifying tool behind information criteria like DIC and WAIC (widely applicable information criterion). Which does not convince me towards the utility of those criteria as model selection tools, as there is too much freedom in the way approximations are used and a potential for using the data several times.

## Posterior predictive p-values and the convex order

Posted in Books, Statistics, University life with tags , , , , , , , , , on December 22, 2014 by xi'an

Patrick Rubin-Delanchy and Daniel Lawson [of Warhammer fame!] recently arXived a paper we had discussed with Patrick when he visited Andrew and I last summer in Paris. The topic is the evaluation of the posterior predictive probability of a larger discrepancy between data and model $\mathbb{P}\left( f(X|\theta)\ge f(x^\text{obs}|\theta) \,|\,x^\text{obs} \right)$

which acts like a Bayesian p-value of sorts. I discussed several times the reservations I have about this notion on this blog… Including running one experiment on the uniformity of the ppp while in Duke last year. One item of those reservations being that it evaluates the posterior probability of an event that does not exist a priori. Which is somewhat connected to the issue of using the data “twice”.

“A posterior predictive p-value has a transparent Bayesian interpretation.”

Another item that was suggested [to me] in the current paper is the difficulty in defining the posterior predictive (pp), for instance by including latent variables $\mathbb{P}\left( f(X,Z|\theta)\ge f(x^\text{obs},Z^\text{obs}|\theta) \,|\,x^\text{obs} \right)\,,$

which reminds me of the multiple possible avatars of the BIC criterion. The question addressed by Rubin-Delanchy and Lawson is how far from the uniform distribution stands this pp when the model is correct. The main result of their paper is that any sub-uniform distribution can be expressed as a particular posterior predictive. The authors also exhibit the distribution that achieves the bound produced by Xiao-Li Meng, Namely that $\mathbb{P}(P\le \alpha) \le 2\alpha$

where P is the above (top) probability. (Hence it is uniform up to a factor 2!) Obviously, the proximity with the upper bound only occurs in a limited number of cases that do not validate the overall use of the ppp. But this is certainly a nice piece of theoretical work. 