The paper deals with designing an approximate active fault detector for stochastic linear Markovian switching systems over an infinite time horizon. The problem is formulated as a functional optimization problem that can be solved using approximate dynamic programming. First, the Generalized Pseudo Bayes (GPB) algorithm is employed to solve the state estimation problem. Then the original formulation is restated by introducing a hyper-state that comprises a finite dimensional statistics obtained from the GPB algorithm. Since the hyper-state is of a higher dimension, a nonparametric local approximation of the Bellman function is used together with the value iteration algorithm to design the approximate active fault detector. The performance of the designed approximate active fault detector is demonstrated through a numerical example.