  Behavioral experiments on humans and animals suggest that the brain performs
probabilistic inference to interpret its environment. Here we present a new
general-purpose, biologically-plausible neural implementation of approximate
inference. The neural network represents uncertainty using Probabilistic
Population Codes (PPCs), which are distributed neural representations that
naturally encode probability distributions, and support marginalization and
evidence integration in a biologically-plausible manner. By connecting multiple
PPCs together as a probabilistic graphical model, we represent multivariate
probability distributions. Approximate inference in graphical models can be
accomplished by message-passing algorithms that disseminate local information
throughout the graph. An attractive and often accurate example of such an
algorithm is Loopy Belief Propagation (LBP), which uses local marginalization
and evidence integration operations to perform approximate inference
efficiently even for complex models. Unfortunately, a subtle feature of LBP
renders it neurally implausible. However, LBP can be elegantly reformulated as
a sequence of Tree-based Reparameterizations (TRP) of the graphical model. We
re-express the TRP updates as a nonlinear dynamical system with both fast and
slow timescales, and show that this produces a neurally plausible solution. By
combining all of these ideas, we show that a network of PPCs can represent
multivariate probability distributions and implement the TRP updates to perform
probabilistic inference. Simulations with Gaussian graphical models demonstrate
that the neural network inference quality is comparable to the direct
evaluation of LBP and robust to noise, and thus provides a promising mechanism
for general probabilistic inference in the population codes of the brain.
