Using the reference probability method and the change of measure in discrete time, the state estimator problem is considered for linear systems observed in Gaussian noise when the coefficients are functions of a noisily observed, finite-state Markov chain. The methods are new, and it is surprising that finite-dimensional filters are obtained. However, the number of statistics increases in time, ii numerical comparison of this filter with the interactive multiple model algorithm introduced by Blom and Bar-Shalom [2] is given.