In many engineering applications the process to be controlled or to be filtered can be modeled as a variable structure system, i.e. a system whose dynamics switches within a finite set of behaviors. Different approaches to the filtering problem can be followed, according to the nature of the switching process. This paper considers the problem of state estimation for discrete-time stochastic systems in which the switching process is modeled as a finite-state Markov Chain. By using a state space realization for the Markov process and the formalism of bilinear systems, a linear filter is developed that achieves the best estimate among all the linear transformations of the measured output. In perspective, the proposed formalism can be used to compute the optimal polynomial state estimates of variable structure systems.