**O**n the flight back from Montpellier, last week, I read a 2019 paper by Gao et al. revisiting the MLE estimation of a parametric family parameter when the normalising constant Z=Z(θ) is unknown. Via noise-contrastive estimation à la Guttman & Hyvärinnen (or à la Charlie Geyer). Treating the normalising constant Z as an extra parameter (as in Kong et al.) and the classification probability as an objective function and calling it a likelihood, which it is not in my opinion as (i) the allocation to the groups is not random and (ii) the original density of the actual observations does not appear in the so-called likelihood.

*“When q appears on the right of KL-divergence* [against *p*],* it is forced to cover most of the modes of p, When q appears on the left of KL-divergence, it tends to chase the major modes of p while ignoring the minor modes.”*

The flow in the title indicates that the contrastive distribution *q* is estimated by a flow-based estimator, namely the transform of a basic noise distribution via easily invertible and differentiable transforms, for instance with lower triangular Jacobians. This flow is also estimated directly from the data but the authors complain this estimation is not good enough for noise contrastive estimation and suggest instead resorting to a GAN version where the classification log-probability is maximised in the model parameters and minimsed in the flow parameters. Except that I feel it misses the true likelihood part. In other words, why on Hyperion would estimating all θ, Z=Z(θ), and α at once improve the estimation of Z?

The other aspect that puzzles me is that (12) uses integrated classification probabilities (with the unknown Z as extra parameter), rather than conditioning on the data, Bayes-like. (The difference between (12) and GAN is that here the discriminator function is constrained.) Esp. when the first expectation is replaced with its empirical version.

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