Markovian Continuity of the MMSE

Minimum mean square error (MMSE) estimation is widely used in signal processing and related fields. While it is known to be non-continuous with respect to all standard notions of stochastic convergence, it remains robust in practical applications. In this work, we review the known counterexamples to the continuity of the MMSE. We observe that, in these counterexamples, the discontinuity arises from an element in the converging measurement sequence providing more information about the estimand than the limit of the measurement sequence. We argue that this behavior is uncharacteristic of real-world applications and introduce a new stochastic convergence notion, termed Markovian convergence, to address this issue. We prove that the MMSE is, in fact, continuous under this new notion. We supplement this result with semi-continuity and continuity guarantees of the MMSE in other settings and prove the continuity of the MMSE under linear estimation.