Parameter Estimation for Gibbs Distributions from Partially Observed Data

Abstract

We study parameter estimation for Markov random fields (MRFs) over $Z^d, d \geq 1$, from incomplete (degraded) data. The MRFs are parameterized by points in a set $\Theta \subseteq \mathbb{R}^m, m \geq 1$. 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 $P^{y \mid x}$. Under natural assumptions on $P^{y \mid x}$, we show that the ML estimations are strongly consistent irrespective of phase transitions, ergodicity or stationarity, provided that $\Theta$ is compact. The same result holds for noncompact $\Theta$ under an extra assumption on the pressure of the MRFs.