Parameter Estimation for Gibbs Distributions from Partially Observed Data

We study parameter estimation for Markov random fields (MRFs) over Zd,d.1, from incomplete (degraded) data. The MRFs are parameterized by points in a set ..Rm,m.1. The interactions are translation invariant but not necessarily of finite range, and the single-pixel random variables take values in a compact space. The observed (degraded) process y takes values in a Polish space, and it is related to the unobserved MRF x via a conditional probability Py.x. Under natural assumptions on Py.x, we show that the ML estimations are strongly consistent irrespective of phase transitions, ergodicity or stationarity, provided that . is compact. The same result holds for noncompact . under an extra assumption on the pressure of the MRFs.