Ising field parameter estimation from incomplete and noisy data

The present paper deals with the estimation problem of the Ising field parameter and extends a previous one [1]. It proposes an estimate from indirect observation (incomplete and noisy), whereas the previous paper proposed an estimate from direct observation (complete and noiseless). Both of them are based on an explicit expression for the partition function, known for a long time [2] but, to the best of our knowledge, never used for parameter estimation (except in our previous paper [1]). Both of them are developed in a Bayesian framework. In our previous study (direct observation), the posterior law is explicit but in the present case (indirect observation) the posterior law is not explicit due to the hidden structure. The proposed approach relies on a full Bayesian strategy and a stochastic sampling algorithm (Gibbs sampler including a Metropolis-Hastings step) for posterior exploration. The paper proposes a numerical evaluation of the proposed method.