  Implicit processes (IPs) are a generalization of Gaussian processes (GPs).
IPs may lack a closed-form expression but are easy to sample from. Examples
include, among others, Bayesian neural networks or neural samplers. IPs can be
used as priors over functions, resulting in flexible models with
well-calibrated prediction uncertainty estimates. Methods based on IPs usually
carry out function-space approximate inference, which overcomes some of the
difficulties of parameter-space approximate inference. Nevertheless, the
approximations employed often limit the expressiveness of the final model,
resulting, e.g., in a Gaussian predictive distribution, which can be
restrictive. We propose here a multi-layer generalization of IPs called the
Deep Variational Implicit process (DVIP). This generalization is similar to
that of deep GPs over GPs, but it is more flexible due to the use of IPs as the
prior distribution over the latent functions. We describe a scalable
variational inference algorithm for training DVIP and show that it outperforms
previous IP-based methods and also deep GPs. We support these claims via
extensive regression and classification experiments. We also evaluate DVIP on
large datasets with up to several million data instances to illustrate its good
scalability and performance.
