In this paper, a novel solution to the active fault diagnosis problem for stochastic linear Markovian switching systems on the infinite-time horizon is proposed. The imperfect state information problem of designing an active fault detector that minimizes a general detection cost criterion is reformulated as the perfect state information problem using sufficient statistics. The reformulation decreases theoretical complexity and enables to find a suboptimal solution by dynamic programming. However, classical approaches are computationally complex or fail to identify the most representative states of the system. This paper combines the active fault detection, state estimation, and reinforcement learning. In the proposed algorithm, temporal difference learning is used to train the active fault detector based on input-output data from the system simulation. The designed detector can be then used online. A numerical example is presented to verify the proposed algorithm.