A Kronecker product model is the set of visible marginal probability
distributions of an exponential family whose sufficient statistics matrix
factorizes as a Kronecker product of two matrices, one for the visible
variables and one for the hidden variables. We estimate the dimension of these
models by the maximum rank of the Jacobian in the limit of large parameters.
The limit is described by the tropical morphism; a piecewise linear map with
pieces corresponding to slicings of the visible matrix by the normal fan of the
hidden matrix. We obtain combinatorial conditions under which the model has the
expected dimension, equal to the minimum of the number of natural parameters
and the dimension of the ambient probability simplex. Additionally, we prove
that the binary restricted Boltzmann machine always has the expected dimension.