Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models

This paper addresses the problem of computing the Cramer-Rao bound for the parameters of a Markov random field. This bound depends on the derivatives of a likelihood distribution that is generally intractable. It is established that by exploiting a property of the exponential family, this intractable bound can be related to the statistical moments of the Gibbs potential of the Markov random field. A derivative-free Monte Carlo algorithm is then proposed to estimate this moments and compute the bound. To illustrate the interest of this method, the proposed algorithm is successfully applied to the Ising and Potts models. The resulting bounds are used to assess the performance of three state-of-the art estimators of the parameter of these Markov random fields.