  We study the natural gradient method for learning in deep Bayesian networks,
including neural networks. There are two natural geometries associated with
such learning systems consisting of visible and hidden units. One geometry is
related to the full system, the other one to the visible sub-system. These two
geometries imply different natural gradients. In a first step, we demonstrate a
great simplification of the natural gradient with respect to the first
geometry, due to locality properties of the Fisher information matrix. This
simplification does not directly translate to a corresponding simplification
with respect to the second geometry. We develop the theory for studying the
relation between the two versions of the natural gradient and outline a method
for the simplification of the natural gradient with respect to the second
geometry based on the first one. This method suggests to incorporate a
recognition model as an auxiliary model for the efficient application of the
natural gradient method in deep networks.
