State estimation algorithms for Markov chains observed in arbitrary noise

In this article we compute state estimation schemes for discrete-time Markov chains observed in arbitrary observation noise. Here we assume the observation noise distribution is known in advance. Appealing to a fundamental L¹ convergence result in[1] we propose to represent any practical observation noise model by a convex combination of Gaussian densities, that is, a mixture function that is itself a valid probability density function. To compute our state estimation schemes we use the techniques of reference probability, (see[2]). Here however, our Gaussian mixtures appear as sums in a product representation of Radon-Nikodym derivatives. The state estimation schemes we compute are; an information state recursion (filter), a general smoothing theorem, an M-ary detection scheme. A computer simulation is provided to indicate the performance of our recursive filter in a non-Gaussian observation noise scenario.