**T**his week, I gave a short and introductory course in Warwick for the CDT (PhD) students on my perceived connections between reverse logistic regression à la Geyer and GANS, among other things. The first attempt was cancelled in 2020 due to the pandemic, the second one in 2021 was on-line and thus offered little possibilities for interactions. Preparing for this third attempt made me read more papers on some statistical analyses of GANs and WGANs, which was more satisfactory [for me] even though I could not get into the technical details…

## Archive for normalising flow

## GANs as density estimators

Posted in Books, Statistics with tags arXiv, Bayesian GANs, conditional sampling, Jacobian, MCMC, neural density estimator, normalising flow, push-forward distribution on October 15, 2021 by xi'an**I** recently read an arXival entitled Conditional Sampling With Monotone GAN by Kovakchi et al., who construct a mapping T that transforms or pushes forward a reference measure þ(dθ) like a multivariate Normal distribution to a target conditional distribution ð(dθ|x). Which makes the proposal a type of normalising flow, except it does not require a Jacobian derivation… The mapping T is monotonous and block triangular in order to be invertible. It is learned from data by minimising a functional divergence between Tþ(dθ) and ð(dθ|x), for instance GAN least square or GAN Wasserstein penalties and representing T as a neural network. Where monotonicity is imposed by a Lagrangian. The authors “note that global minimizers of [their GAN criterion] can also be used for conditional density estimation” but I fail to understand the distinction in that once T is constructed, the estimated conditional density is automatically available. However my main source of puzzlement is at the worth of this construction, since it does not provide an exact generative process for the conditional distribution, while requiring many generations from the joint distribution. Rather than a comparison with MCMC, which is not applicable in untractable generative models, a comparison with less expensive ABC solutions would have been appropriate, I think. And the paper is missing any quantification on the quality or asymptotics of the density estimate provided by this involved approximation, as most of the recent literature on normalising flows and friends. (A point acknowledged by the authors in the supplementary material section.)

“In this regard, the MGANs approach introduced in the article belongs to the category of sampling techniques such as MCMC, whose goal is to generate independent samples from the law of y|x, as opposed to assuming some structural form of the probability measure directly.”

I am unsure I understand the above remark as MCMC methods are intrinsically linked with the exact probability distribution, exploiting either some conditional representations as in Gibbs or at the very least the ability to compute the joint density…

## improving bridge samplers by GANs

Posted in Books, pictures, Statistics with tags bridge sampling, curse of dimensionality, GANs, noise contrasting estimation, normalising flow, PhD, Saint Giles cemetery, University of Oxford on July 20, 2021 by xi'an**H**anwen Xing from Oxford recently posted a paper on arXiv about using GANs to improve the overlap bewtween the densities in bridge sampling. Bringing out new connections with noise contrastive estimation. The idea is to optimise a transform of one of the densities *h()* to bring it closer to the other density *k()*, using for instance normalising flows. (The call to transforms for bridge is not new, dating at least to Voter in 1985, the year I was starting my PhD!) Furthermore, using an f-divergence as a measure of functional distance allows for a reasonably straightforward update of the transform. That can be reformulated as a GAN target, which is somewhat natural in that the transform aims at confusing simulation from the transform of *h* and from *k*. This is quite an interesting proposal, even though calculating the optimal transform is time-consuming and subjet to the curse of dimensionality. I also wonder at whether or not iterating the optimisation, one density after the other, would be bring further improvement.

## more air for MCMC

Posted in Books, R, Statistics with tags ANOVA models, Bayesian Analysis, Bayesian deep learning, BUGS, coda, convergence assessment, convergence diagnostics, ESS, MCMC convergence, multiple chains, normalising flow, p-values, R factor, STAN, Valencia conferences on May 30, 2021 by xi'an**A**ki Vehtari, Andrew Gelman, Dan Simpson, Bob Carpenter, and Paul-Christian Bürkner have just published a Bayesian Analysis paper about using an improved R factor for MCMC convergence assessment. From the early days of MCMC, convergence assessment has been a recurring (and recurrent!) question in the community. First leading to a flurry of proposals, [which Kerrie, Chantal, and myself reviewwwed in the Valencia 1998 proceedings], and then slowly disintegrating under the onslaughts of reality—i.e. that none could not be 100% foolproof in full generality—…. This included the (possibly now forgotten) *single-versus-multiple-chains* debate between Charlie Geyer [for single] and Andrew Gelman and Don Rubin [for multiple]. The later introduced an analysis-of-variance R factor, which remains quite popular up to this day, in part for being part of most MCMC software, like BUGS. That this R may fail to identify convergence issues, even in the more recent split version, does not come as a major surprise, since any situation with a long-term influence of the starting distribution may well fail to identify missing (significant) parts of the posterior support. (It is thus somewhat disconcerting to me to see that the main recommendation is to move the bound on R from 1.1 to 1.01, reminding me to some extent of a recent proposal to move the null rejection boundary from 0.05 to 0.005…) Similarly, the ESS may prove a poor signal for convergence or lack thereof, especially because the approximation of the asymptotic variance relies on stationarity assumptions. While multiplying the monitoring tools (as in CODA) helps with identifying convergence issues, looking at a single convergence indicator is somewhat like looking only at a frequentist estimator! (And with greater automation comes greater responsibility—in keeping a critical perspective.)

Looking for a broader perspective, I thus wonder at what we would instead need to assess the lack of convergence of an MCMC chain without much massaging of the said chain. An evaluation of the (Kullback, Wasserstein, or else) distance between the distribution of the chain at iteration n or across iterations, and the true target? A percentage of the mass of the posterior visited so far, which relates to estimating the normalising constant, with a relatively vast array of solutions made available in the recent years? I remain perplexed and frustrated by the fact that, 30 years later, the computed values of the visited likelihoods are not better exploited. Through for instance machine-learning approximations of the target. that could themselves be utilised for approximating the normalising constant and potential divergences from other approximations.

## NCE, VAEs, GANs & even ABC…

Posted in Statistics with tags ABC, Bayesian GANs, CDT, deep learning, energy based model, generative adversarial networks, noise contrasting estimation, normalising constant, normalising flow, partition function, PhD course, Teams, University of Warwick, variational autoencoders on May 14, 2021 by xi'an**A**s I was preparing my (new) lectures for a PhD short course “at” Warwick (meaning on Teams!), I read a few surveys and other papers on all these acronyms. It included the massive Guttmann and Hyvärinen 2012 NCE JMLR paper, Goodfellow’s NIPS 2016 tutorial on GANs, and Kingma and Welling 2019 introduction to VAEs. Which I found a wee bit on the light side, maybe missing the fundamentals of the notion… As well as the pretty helpful 2019 survey on normalising flows by Papamakarios et al., although missing on the (statistical) density estimation side. And also a nice (2017) survey of GANs by Shakir Mohamed and Balaji Lakshminarayanan with a somewhat statistical spirit, even though convergence issues are not again not covered. But misspecification is there. And the many connections between ABC and GANs, if definitely missing on the uncertainty aspects. While Deep Learning by Goodfellow, Bengio and Courville adresses both the normalising constant (or partition function) and GANs, it was somehow not deep enough (!) to use for the course, offering only a few pages on NCE, VAEs and GANs. (And also missing on the statistical references addressing the issue, incl. [or excl.] Geyer, 1994.) Overall, the infinite variations offered on GANs leave me uncertain about their statistical relevance, as it is unclear how good the regularisation therein is for handling overfitting and consistent estimation. (And if I spot another decomposition of the Kullback-Leibler divergence, I may start crying…)