Implementing Bayesian Inference with Neural Networks

Embodied agents, be they animals or robots, acquire information about the world through their senses. Embodied agents, however, do not simply lose this information once it passes by, but rather process and store it for future use. The most general theory of how an agent can combine stored knowledge with new observations is Bayesian inference. In this dissertation I present a theory of how embodied agents can learn to implement Bayesian inference with neural networks. By neural network I mean both artificial and biological neural networks, and in my dissertation I address both kinds. On one hand, I develop theory for implementing Bayesian inference in deep generative models, and I show how to train multilayer perceptrons to compute approximate predictions for Bayesian filtering. On the other hand, I show that several models in computational neuroscience are special cases of the general theory that I develop in this dissertation, and I use this theory to model and explain several phenomena in neuroscience. The key contributions of this dissertation can be summarized as follows: - I develop a class of graphical model called nth-order harmoniums. An nth-order harmonium is an n-tuple of random variables, where the conditional distribution of each variable given all the others is always an element of the same exponential family. I show that harmoniums have a recursive structure which allows them to be analyzed at coarser and finer levels of detail. - I define a class of harmoniums called rectified harmoniums, which are constrained to have priors which are conjugate to their posteriors. As a consequence of this, rectified harmoniums afford efficient sampling and learning. - I develop deep harmoniums, which are harmoniums which can be represented by hierarchical, undirected graphs. I develop the theory of rectification for deep harmoniums, and develop a novel algorithm for training deep generative models. - I show how to implement a variety of optimal and near-optimal Bayes