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On the total variation distance of labelled Markov chains

Stefan Kiefer ( University of Oxford )

Labelled Markov chains (LMCs) are widely used in probabilistic verification, speech recognition, computational biology, and many other fields. Checking two LMCs for equivalence is a classical problem subject to extensive studies, while the total variation distance provides a natural measure for the 'inequivalence' of two LMCs: it is the maximum difference between probabilities that the LMCs assign to the same event. In this talk we develop a theory of the total variation distance between two LMCs, with emphasis on the algorithmic aspects: (1) we provide a polynomial-time algorithm for determining whether two LMCs have distance 1, i.e., whether they can almost always be distinguished; (2) we provide an algorithm for approximating the distance with arbitrary precision; and (3) we show that the threshold problem, i.e., whether the distance exceeds a given threshold, is NP-hard and hard for the square-root-sum problem. We also make a connection between the total variation distance and Bernoulli convolutions.

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