  Deep belief networks are a powerful way to model complex probability
distributions. However, learning the structure of a belief network,
particularly one with hidden units, is difficult. The Indian buffet process has
been used as a nonparametric Bayesian prior on the directed structure of a
belief network with a single infinitely wide hidden layer. In this paper, we
introduce the cascading Indian buffet process (CIBP), which provides a
nonparametric prior on the structure of a layered, directed belief network that
is unbounded in both depth and width, yet allows tractable inference. We use
the CIBP prior with the nonlinear Gaussian belief network so each unit can
additionally vary its behavior between discrete and continuous representations.
We provide Markov chain Monte Carlo algorithms for inference in these belief
networks and explore the structures learned on several image data sets.
