Parameter Estimation for Gibbs Distributions from Partially Observed Data

We study parameter estimation for Markov Random Fields (MRF) over Z d from incomplete data. The MRF are parametrized by points in a set Θ⊂R m . 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 process y takes values in a Polish space, and it is related to the unobserved MRF x via a conditional probability. Under natural assumptions on this probability, we show that the ML estimations are strongly consistent irrespectively of phase transitions, ergodicity, or stationarity, provided that Θ is compact