## GANs as density estimators

**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…

October 16, 2021 at 10:05 pm

“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.”

Under the circumstances, i.e. accessing \nu only via forward simulation, I would think that an exact generative process for the conditional distribution is a pretty big ask? (unless you meant an exact generative process for their approximate conditional distribution, which is given implicitly in Theorem 1)