The problem of anomaly detection among multiple processes is considered within the framework of sequential design of experiments. The objective is an active inference strategy consisting of a selection rule governing which process to probe at each time, a stopping rule on when to terminate the detection, and a decision rule on the final detection outcome. The performance measure is the Bayes risk that takes into account not only sample complexity and detection errors, but also costs associated with switching across processes. While the problem is a partially observable Markov decision process to which optimal solutions are generally intractable, a low-complexity deterministic policy is shown to be asymptotically optimal and offer significant performance improvement over existing methods in the finite regime.