Mixtures, Markov Bridges and the Matrix Product Ansatz

We give a probabilistic characterization of the set of measures that can be represented by the Matrix Product Ansatz. By suitably enlarging the state space, we show that a probability measure can be described in terms of non-negative matrices by the Matrix Product Ansatz, if and only if it can be written as a mixture of inhomogeneous product measures where the mixing law is a Markov bridge. We give a constructive procedure to identify such probabilistic features. We illustrate the result by examples and show that existing probabilistic representations of the invariant measures of non-equilibrium interacting particle systems can be obtained from the matrix product ansatz by this general procedure.