  In this report a derivation of the MAP state estimator objective function for
general (possibly non-square) discrete time causal/non-causal descriptor
systems is presented. The derivation made use of the Kronecker Canonical
Transformation to extract the prior distribution on the descriptor state vector
so that Maximum a Posteriori (MAP) point estimation can be used. The analysis
indicates that the MAP estimate for index 1 causal descriptor systems does not
require any model transformations and can be found recursively. Furthermore, if
the descriptor system is of index 2 or higher and the noise free system is
causal, then the MAP estimate can also be found recursively without model
transformations provided that model causality is accounted for in designing the
stochastic model.
