  A finite dimensional abstract approximation and convergence theory is
developed for estimation of the distribution of random parameters in infinite
dimensional discrete time linear systems with dynamics described by regularly
dissipative operators and involving, in general, unbounded input and output
operators. By taking expectations, the system is re-cast as an equivalent
abstract parabolic system in a Gelfand triple of Bochner spaces wherein the
random parameters become new space-like variables. Estimating their
distribution is now analogous to estimating a spatially varying coefficient in
a standard deterministic parabolic system. The estimation problems are
approximated by a sequence of finite dimensional problems. Convergence is
established using a state space-varying version of the Trotter-Kato semigroup
approximation theorem. Numerical results for a number of examples involving the
estimation of exponential families of densities for random parameters in a
diffusion equation with boundary input and output are presented and discussed.
