training energy based models

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.

on completeness

Another X validated question that proved a bit of a challenge, enough for my returning to its resolution on consecutive days. The question was about the completeness of the natural sufficient statistic associated with a sample from the shifted exponential distribution

[weirdly called negative exponential in the question] meaning the (minimal) sufficient statistic is made of the first order statistic and of the sample sum (or average), or equivalently

Finding the joint distribution of T is rather straightforward as the first component is a drifted Exponential again and the second a Gamma variate with n-2 degrees of freedom and the scale θ². (Devroye’s Bible can be invoked since the Gamma distribution follows from his section on Exponential spacings, p.211.) While the derivation of a function with constant expectation is straightforward for the alternate exponential distribution

since the ratio of the components of T has a fixed distribution, it proved harder for the current case as I was seeking a parameter free transform. When attempting to explain the difficulty on my office board, I realised I was seeking the wrong property since an expectation was enough. Removing the dependence on θ was simpler and led to

but one version of a transform with fixed expectation. This also led me to wonder at the range of possible functions of θ one could use as scale and still retrieve incompleteness of T. Any power of θ should work but what about exp(θ²) or sin²(θ³), i.e. functions for which there exists no unbiased estimator..?

absurdly unbiased estimators

“…there are important classes of problems for which the mathematics forces the existence of such estimators.”

Recently I came through a short paper written by Erich Lehmann for The American Statistician, Estimation with Inadequate Information. He analyses the apparent absurdity of using unbiased estimators or even best unbiased estimators in settings like the Poisson P(λ) observation X producing the (unique) unbiased estimator of exp(-bλ) equal to

which is indeed absurd when b>1. My first reaction to this example is that the question of what is “best” for a single observation is not very meaningful and that adding n independent Poisson observations replaces b with b/n, which gets eventually less than one. But Lehmann argues that the paradox stems from a case of missing information, as for instance in the Poisson example where the above quantity is the probability P(T=0) that T=0, when T=X+Y, Y being another unobserved Poisson with parameter (b-1)λ. In a lot of such cases, there is no unbiased estimator at all. When there is any, it must take values outside the (0,1) range, thanks to a lemma shown by Lehmann that the conditional expectation of this estimator given T is either zero or one.

I find the short paper quite interesting in exposing some reasons why the estimators cannot find enough information within the data (often a single point) to achieve an efficient estimation of the targeted function of the parameter, even though the setting may appear rather artificial.

unbiased estimation of log-normalising constants

Maxime Rischard, Pierre Jacob, and Natesh Pillai [warning: both of whom are co-authors and friends of mine!] have just arXived a paper on the use of path sampling (a.k.a., thermodynamic integration) for log-constant unbiased approximation and the resulting consequences on Bayesian model comparison by X validation. If the goal is the estimation of the log of a ratio of two constants, creating an artificial path between the corresponding distributions and looking at the derivative at any point of this path of the log-density produces an unbiased estimator. Meaning that random sampling along the path, corrected by the distribution of the sampling still produces an unbiased estimator. From there the authors derive an unbiased estimator for any X validation objective function, CV(V,T)=-log p(V|T), taking m observations T in and leaving n-m observations T out… The marginal conditional log density in the criterion is indeed estimated by an unbiased path sampler, using a powered conditional likelihood. And unbiased MCMC schemes à la Jacob et al. for simulating unbiased MCMC realisations of the intermediary targets on the path. Tuning it towards an approximately constant cost for all powers.

So in all objectivity and fairness (!!!), I am quite excited by this new proposal within my favourite area! Or rather two areas since it brings together the estimation of constants and an alternative to Bayes factors for Bayesian testing. (Although the paper does not broach upon the calibration of the X validation values.)

Bayesian synthetic likelihood

Leah Price, Chris Drovandi, Anthony Lee and David Nott published earlier this year a paper in JCGS on Bayesian synthetic likelihood, using Simon Wood’s synthetic likelihood as a substitute to the exact likelihood within a Bayesian approach. While not investigating the theoretical properties of this approximate approach, the paper compares it with ABC on some examples. In particular with respect to the number n of Monte Carlo replications used to approximate the mean and variance of the Gaussian synthetic likelihood.

Since this approach is most naturally associated with an MCMC implementation, it requires new simulations of the summary statistics at each iteration, without a clear possibility to involve parallel runs, in contrast to ABC. However in the final example of the paper, the authors reach values of n of several thousands, making use of multiple cores relevant, if requiring synchronicity and checks at every MCMC iteration.

The authors mention that “ABC can be viewed as a pseudo-marginal method”, but this has a limited appeal since the pseudo-marginal is a Monte Carlo substitute for the ABC target, not the original target. Similarly, there exists an unbiased estimator of the Gaussian density due to Ghurye and Olkin (1969) that allows to perceive the estimated synthetic likelihood version as a pseudo-marginal, once again wrt a target that differs from the original one. And the bias reappears under mis-specification, that is when the summary statistics are not normally distributed. It seems difficult to assess this normality or absence thereof in realistic situations.

“However, when the distribution of the summary statistic is highly irregular, the output of BSL cannot be trusted, while ABC represents a robust alternative in such cases.”

To make synthetic likelihood and ABC algorithms compatible, the authors chose a Normal kernel for ABC. Still, the equivalence is imperfect in that the covariance matrix need be chosen in the ABC case and is estimated in the synthetic one. I am also lost to the argument that the synthetic version is more efficient than ABC, in general (page 8). As for the examples, the first one uses a toy Poisson posterior with a single sufficient summary statistic, which is not very representative of complex situations where summary statistics are extremes or discrete. As acknowledged by the authors this is a case when the Normality assumption applies. For an integer support hidden process like the Ricker model, normality vanishes and the outcomes of ABC and synthetic likelihood differ, which makes it difficult to compare the inferential properties of both versions (rather than the acceptance rates), while using a 13-dimension statistic for estimating a 3-dimension parameter is not recommended for ABC, as discussed by Li and Fearnhead (2017). The same issue appears in the realistic cell motility example, with 145 summaries versus two parameters. (In the philogenies studied by DIYABC, the number of summary statistics is about the same but we now advocate a projection to the parameter dimension by the medium of random forests.)

Given the similarity between both approaches, I wonder at a confluence between them, where synthetic likelihood could maybe be used to devise PCA on the summary statistics and facilitate their projection on a space with much smaller dimensions. Or estimating the mean and variance functions in the synthetic likelihood towards producing directly simulations of the summary statistics.