Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for models which encompass our understanding of the physical word. Despite this fundamental nature, the use of implicit models remains limited due to challenge in positing complex latent structure in them, and the ability to inference in such models with large data sets. In this paper, we first introduce the 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 symbol 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.

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for models which encompass our understanding of the physical word. Despite this fundamental nature, the use of implicit models remains limited due to challenge in positing complex latent structure in them, and the ability to inference in such models with large data sets. In this paper, we first introduce the 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.