  Most causal discovery algorithms find causal structure among a set of
observed variables. Learning the causal structure among latent variables
remains an important open problem, particularly when using high-dimensional
data. In this paper, we address a problem for which it is known that inputs
cause outputs, and these causal relationships are encoded by a causal network
among a set of an unknown number of latent variables. We developed a deep
learning model, which we call a redundant input neural network (RINN), with a
modified architecture and a regularized objective function to find causal
relationships between input, hidden, and output variables. More specifically,
our model allows input variables to directly interact with all latent variables
in a neural network to influence what information the latent variables should
encode in order to generate the output variables accurately. In this setting,
the direct connections between input and latent variables makes the latent
variables partially interpretable; furthermore, the connectivity among the
latent variables in the neural network serves to model their potential causal
relationships to each other and to the output variables. A series of simulation
experiments provide support that the RINN method can successfully recover
latent causal structure between input and output variables.
