  A key goal of unsupervised learning is to go beyond density estimation and
sample generation to reveal the structure inherent within observed data. Such
structure can be expressed in the pattern of interactions between explanatory
latent variables captured through a probabilistic graphical model. Although the
learning of structured graphical models has a long history, much recent work in
unsupervised modelling has instead emphasised flexible deep-network-based
generation, either transforming independent latent generators to model complex
data or assuming that distinct observed variables are derived from different
latent nodes. Here, we extend amortised variational inference to incorporate
structured factors over multiple variables, able to capture the
observation-induced posterior dependence between latents that results from
``explaining away'' and thus allow complex observations to depend on multiple
nodes of a structured graph. We show that appropriately parametrised factors
can be combined efficiently with variational message passing in rich graphical
structures. We instantiate the framework in nonlinear Gaussian Process Factor
Analysis, evaluating the structured recognition framework using synthetic data
from known generative processes. We fit the GPFA model to high-dimensional
neural spike data from the hippocampus of freely moving rodents, where the
model successfully identifies latent signals that correlate with behavioural
covariates.
