Archive for Hyvärinen score

training energy based models

Posted in Books, Statistics with tags , , , , , , , on April 7, 2021 by xi'an

This recent arXival by Song and Kingma covers different computational approaches to semi-parametric estimation, but also exposes imho the chasm existing between statistical and machine learning perspectives on the problem.

“Energy-based models are much less restrictive in functional form: instead of specifying a normalized probability, they only specify the unnormalized negative log-probability (…) Since the energy function does not need to integrate to one, it can be parameterized with any nonlinear regression function.”

The above in the introduction appears first as a strange argument, since the mass one constraint is the least of the problems when addressing non-parametric density estimation. Problems like the convergence, the speed of convergence, the computational cost and the overall integrability of the estimator. It seems however that the restriction or lack thereof is to be understood as the ability to use much more elaborate forms of densities, which are then black-boxes whose components have little relevance… When using such mega-over-parameterised representations of densities, such as neural networks and normalising flows, a statistical assessment leads to highly challenging questions. But convergence (in the sample size) does not appear to be a concern for the paper. (Except for a citation of Hyvärinen on p.5.)

Using MLE in this context appears to be questionable, though, since the base parameter θ is not unlikely to remain identifiable. Computing the MLE is therefore a minor issue, in this regard, a resolution based on simulated gradients being well-chartered from the earlier era of stochastic optimisation as in Robbins & Monro (1954), Duflo (1996) or Benveniste & al. (1990). (The log-gradient of the normalising constant being estimated by the opposite of the gradient of the energy at a random point.)

“Running MCMC till convergence to obtain a sample x∼p(x) can be computationally expensive.”

Contrastive divergence à la Hinton (2002) is presented as a solution to the convergence problem by stopping early, which seems reasonable given the random gradient is mostly noise. With a possible correction for bias à la Jacob & al. (missing the published version).

An alternative to MLE is the 2005 Hyvärinen score, notorious for bypassing the normalising constant. But blamed in the paper for being costly in the dimension d of the variate x, due to the second derivative matrix. Which can be avoided by using Stein’s unbiased estimator of the risk (yay!) if using randomized data. And surprisingly linked with contrastive divergence as well, if a Taylor expansion is good enough an approximation! An interesting byproduct of the discussion on score matching is to turn it into an unintended form of ABC!

“Many methods have been proposed to automatically tune the noise distribution, such as Adversarial Contrastive Estimation (Bose et al., 2018), Conditional NCE (Ceylan and Gutmann, 2018) and Flow Contrastive Estimation (Gao et al., 2020).”

A third approach is the noise contrastive estimation method of Gutmann & Hyvärinen (2010) that connects with both others. And is a precursor of GAN methods, mentioned at the end of the paper via a (sort of) variational inequality.

Siem Reap conference

Posted in Kids, pictures, Travel, University life with tags , , , , , , , , , , , , , , , , , , on March 8, 2019 by xi'an

As I returned from the conference in Siem Reap. on a flight avoiding India and Pakistan and their [brittle and bristling!] boundary on the way back, instead flying far far north, near Arkhangelsk (but with nothing to show for it, as the flight back was fully in the dark), I reflected how enjoyable this conference had been, within a highly friendly atmosphere, meeting again with many old friends (some met prior to the creation of CREST) and new ones, a pleasure not hindered by the fabulous location near Angkor of course. (The above picture is the “last hour” group picture, missing a major part of the participants, already gone!)

Among the many talks, Stéphane Shao gave a great presentation on a paper [to appear in JASA] jointly written with Pierre Jacob, Jie Ding, and Vahid Tarokh on the Hyvärinen score and its use for Bayesian model choice, with a highly intuitive representation of this divergence function (which I first met in Padua when Phil Dawid gave a talk on this approach to Bayesian model comparison). Which is based on the use of a divergence function based on the squared error difference between the gradients of the true log-score and of the model log-score functions. Providing an alternative to the Bayes factor that can be shown to be consistent, even for some non-iid data, with some gains in the experiments represented by the above graph.

Arnak Dalalyan (CREST) presented a paper written with Lionel Riou-Durand on the convergence of non-Metropolised Langevin Monte Carlo methods, with a new discretization which leads to a substantial improvement of the upper bound on the sampling error rate measured in Wasserstein distance. Moving from p/ε to √p/√ε in the requested number of steps when p is the dimension and ε the target precision, for smooth and strongly log-concave targets.

This post gives me the opportunity to advertise for the NGO Sala Baï hostelry school, which the whole conference visited for lunch and which trains youths from underprivileged backgrounds towards jobs in hostelery, supported by donations, companies (like Krama Krama), or visiting the Sala Baï  restaurant and/or hotel while in Siem Reap.

 

Implicit maximum likelihood estimates

Posted in Statistics with tags , , , , , , , , , , on October 9, 2018 by xi'an

An ‘Og’s reader pointed me to this paper by Li and Malik, which made it to arXiv after not making it to NIPS. While the NIPS reviews were not particularly informative and strongly discordant, the authors point out in the comments that they are available for the sake of promoting discussion. (As made clear in earlier posts, I am quite supportive of this attitude! Disclaimer: I was not involved in an evaluation of this paper, neither for NIPS nor for another conference or journal!!) Although the paper does not seem to mention ABC in the setting of implicit likelihoods and generative models, there is a reference to the early (1984) paper by Peter Diggle and Richard Gratton that is often seen as the ancestor of ABC methods. The authors point out numerous issues with solutions proposed for parameter estimation in such implicit models. For instance, for GANs, they signal that “minimizing the Jensen-Shannon divergence or the Wasserstein distance between the empirical data distribution and the model distribution does not necessarily minimize the same between the true data distribution and the model distribution.” (Not mentioning the particular difficulty with Bayesian GANs.) Their own solution is the implicit maximum likelihood estimator, which picks the value of the parameter θ bringing a simulated sample the closest to the observed sample. Closest in the sense of the Euclidean distance between both samples. Or between the minimum of several simulated samples and the observed sample. (The modelling seems to imply the availability of n>1 observed samples.) They advocate using a stochastic gradient descent approach for finding the optimal parameter θ which presupposes that the dependence between θ and the simulated samples is somewhat differentiable. (And this does not account for using a min, which would make differentiation close to impossible.) The paper then meanders in a lengthy discussion as to whether maximising the likelihood makes sense, with a rather naïve view on why using the empirical distribution in a Kullback-Leibler divergence does not make sense! What does not make sense is considering the finite sample approximation to the Kullback-Leibler divergence with the true distribution in my opinion.

Au’Bayes 17

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on December 14, 2017 by xi'an

Some notes scribbled during the O’Bayes 17 conference in Austin, not reflecting on the highly diverse range of talks. And many new faces and topics, meaning O’Bayes is alive and evolving. With all possible objectivity, a fantastic conference! (Not even mentioning the bars where Peter Müller hosted the poster sessions, a feat I would have loved to see duplicated for the posters of ISBA 2018… Or the Ethiopian restaurant just around the corner with the right amount of fierce spices!)

The wiki on objective, reference, vague, neutral [or whichever label one favours] priors that was suggested at the previous O’Bayes meeting in Valencià, was introduced as Wikiprevia by Gonzalo Garcia-Donato. It aims at classifying recommended priors in most of the classical models, along with discussion panels, and it should soon get an official launch, when contributors will be welcome to include articles in a wiki principle. I wish the best to this venture which, I hope, will induce O’Bayesians to contribute actively.

In a brilliant talk that quickly reverted my jetlag doziness, Peter Grünwald returned to the topic he presented last year in Sardinia, namely safe Bayes or powered-down likelihoods to handle some degree of misspecification, with a further twist of introducing an impossible value `o’ that captures missing mass (to be called Peter’s demon?!), which absolute necessity I did not perceive. Food for thoughts, definitely. (But I feel that the only safe Bayes is the dead Bayes, as protecting against all kinds of mispecifications means no action is possible.)

I also appreciated Cristiano Villa’s approach to constructing prior weights in model comparison from a principled and decision-theoretic perspective even though I felt that the notion of ranking parameter importance required too much input to be practically feasible. (Unless I missed that point.)

Laura Ventura gave her talk on using for ABC various scores or estimating equations as summary statistics, rather than the corresponding M-estimators, which offers the appealing feature of reducing computation while being asymptotically equivalent. (A feature we also exploited for the regular score function in our ABC paper with Gael, David, Brendan, and Wonapree.) She mentioned the Hyvärinen score [of which I first heard in Padova!] as a way to bypass issues related to doubly intractable likelihoods. Which is a most interesting proposal that bypasses (ABC) simulations from such complex targets by exploiting a pseudo-posterior.

Veronika Rockova presented a recent work on concentration rates for regression tree methods that produce a rigorous analysis of these methods. Showing that the spike & slab priors plus BART [equals spike & tree] achieve sparsity and optimal concentration. In an oracle sense. With a side entry on assembling partition trees towards creating a new form of BART. Which made me wonder whether or not this was also applicable to random forests. Although they are not exactly Bayes. Demanding work in terms of the theory behind but with impressive consequences!

Just before I left O’Bayes 17 for Houston airport, Nick Polson, along with Peter McCullach, proposed an intriguing notion of sparse Bayes factors, which corresponds to the limit of a Bayes factor when the prior probability υ of the null goes to zero. When the limiting prior is replaced with an exceedance measure that can be normalised into a distribution, but does it make the limit a special prior? Linking  υ with the prior under the null is not an issue (this was the basis of my 1992 Lindley paradox paper) but the sequence of priors indexed by υ need be chosen. And reading from the paper at Houston airport, I could not spot a construction principle that would lead to a reference prior of sorts. One thing that Nick mentioned during his talk was that we observed directly realisations of the data marginal, but this is generally not the case as the observations are associated with a given value of the parameter, not one for each observation.The next edition of the O’Bayes conference will be in… Warwick on June 29-July 2, as I volunteered to organise this edition (16 years after O’Bayes 03 in Aussois!) just after the BNP meeting in Oxford on June 23-28, hopefully creating the environment for fruitful interactions between both communities! (And jumping from Au’Bayes to Wa’Bayes.)

the Hyvärinen score is back

Posted in pictures, Statistics, Travel with tags , , , , , , , , , , , , , on November 21, 2017 by xi'an

Stéphane Shao, Pierre Jacob and co-authors from Harvard have just posted on arXiv a new paper on Bayesian model comparison using the Hyvärinen score

\mathcal{H}(y, p) = 2\Delta_y \log p(y) + ||\nabla_y \log p(y)||^2

which thus uses the Laplacian as a natural and normalisation-free penalisation for the score test. (Score that I first met in Padova, a few weeks before moving from X to IX.) Which brings a decision-theoretic alternative to the Bayes factor and which delivers a coherent answer when using improper priors. Thus a very appealing proposal in my (biased) opinion! The paper is mostly computational in that it proposes SMC and SMC² solutions to handle the estimation of the Hyvärinen score for models with tractable likelihoods and tractable completed likelihoods, respectively. (Reminding me that Pierre worked on SMC² algorithms quite early during his Ph.D. thesis.)

A most interesting remark in the paper is to recall that the Hyvärinen score associated with a generic model on a series must be the prequential (predictive) version

\mathcal{H}_T (M) = \sum_{t=1}^T \mathcal{H}(y_t; p_M(dy_t|y_{1:(t-1)}))

rather than the version on the joint marginal density of the whole series. (Followed by a remark within the remark that the logarithm scoring rule does not make for this distinction. And I had to write down the cascading representation

\log p(y_{1:T})=\sum_{t=1}^T \log p(y_t|y_{1:t-1})

to convince myself that this unnatural decomposition, where the posterior on θ varies on each terms, is true!) For consistency reasons.

This prequential decomposition is however a plus in terms of computation when resorting to sequential Monte Carlo. Since each time step produces an evaluation of the associated marginal. In the case of state space models, another decomposition of the authors, based on measurement densities and partial conditional expectations of the latent states allows for another (SMC²) approximation. The paper also establishes that for non-nested models, the Hyvärinen score as a model selection tool asymptotically selects the closest model to the data generating process. For the divergence induced by the score. Even for state-space models, under some technical assumptions.  From this asymptotic perspective, the paper exhibits an example where the Bayes factor and the Hyvärinen factor disagree, even asymptotically in the number of observations, about which mis-specified model to select. And last but not least the authors propose and assess a discrete alternative relying on finite differences instead of derivatives. Which remains a proper scoring rule.

I am quite excited by this work (call me biased!) and I hope it can induce following works as a viable alternative to Bayes factors, if only for being more robust to the [unspecified] impact of the prior tails. As in the above picture where some realisations of the SMC² output and of the sequential decision process see the wrong model being almost acceptable for quite a long while…