Hierarchical Implicit Models and Likelihood-Free Variational Inference
Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.
The paper defines a class of probability models -- hierarchical
implicit models -- consisting of observations with associated 'local'
latent variables that are conditionally independent given a set of
'global' latent variables, and in which the observation likelihood is
not assumed to be tractable. It describes an approach for KL-based
variational inference in such 'likelihood-free' models, using a
GAN-style discriminator to estimate the log ratio between a
'variational joint' q(x, z), constructed using the empirical
distribution on observations, and the true model joint density. This
approach has the side benefit of supporting implicit variational
models ('variational programs'). Proof-of-concept applications are
demonstrated to ecological simulation, a Bayesian GAN, and sequence
modeling with a stochastic RNN.
The exposition is very clear, well cited, and the technical machinery
is carefully explained. Although the the application of density ratio
estimation to variational inference seems to be an idea 'in the air'
and building blocks of this paper have appeared elsewhere (for example
the Adversarial VB paper), I found this synthesis to be cleaner,
easier to follow, and more general (supporting implicit models) than
any of the similar papers I've read so far.
The definition of hierarchical implicit models is a useful point in
theoretical space, and serves to introduce the setup for inference
in section 3. However the factorization (1), which assumes iid
observations, is quite restrictive -- I don't believe it technically
even includes the Lotka-Volterra or stochastic RNN models explored in
the paper itself! (since both have temporal dependence). It seems
worth acknowledging that the inference approach in this paper is more
general, and perhaps discussing how it could be adapted to problems
and models with more structured (time series, text, graph)
observations and/or latents.
Experiments are probably the weakest point of this paper. The 'Bayesian GAN' 
is a toy and the classification setup is artificial; supervised learning is
not why people care about GANs. The symbol generation RNN is not
evaluated against any other methods and it's not clear it works
particularly well. The Lotka-Volterra simulation is the most
compelling; although the model has few parameters and no latent
variables, it nicely motivates the notion of implicit models and shows
clear improvement on the (ABC) state of the art.
Overall there are no groundbreaking results, and much of this
machinery could be quite tricky to get working in practice (as with
vanilla GANS). I wish the experiments were more compelling. 
But the approach seems general and powerful, with the
potential to open up entire new classes of models to effective
Bayesian inference, and the formulation in this paper will likely be
useful to many reasearchers as they begin to flesh it out. For that
reason I think this paper is a valuable contribution.
Misc comments and questions:
Lotka-Volterra model: I'm not sure the given eqns (ln 103) are
correct. Shouldn't the Beta_3 be added, not subtracted, to model the
predator birth rate? As written, dx_2/dt is always negative in
expectation which seems wrong. Also Beta_2 is serving double duty as
the predator *and* prey death rate, is this intentional? Most sources
(including the cited Papamakarios and Murray paper) seem to use four
independent coefficients.
line 118: "We described two classes of implicit models" but I only see
one? (HIMs)
line 146: "log empirical log q(x_n)" is redundant
Suppose we have an implicit model, but want to use an explicit
variational approximation (for example the mean-field Gaussian in the
Lotka-Volterra experiment). Is there any natural way to exploit the
explicit variational density for faster inference?
Subtracting the constant log q(x) from the ELBO means the ratio
objective (4) no longer yields a lower bound to the true model
evidence; this should probably be noted somewhere. Is there an
principled interpretation of the quantity (4)? It is a lower bound on
log p(x)/q(x), which (waving hands) looks like an estimate of the
negative KL divergence between the model and empirical distribution --
maybe this is useful for model criticism?