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February, 1992 Parameter Estimation for Gibbs Distributions from Partially Observed Data
Francis Comets, Basilis Gidas
Ann. Appl. Probab. 2(1): 142-170 (February, 1992). DOI: 10.1214/aoap/1177005775


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.


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Francis Comets. Basilis Gidas. "Parameter Estimation for Gibbs Distributions from Partially Observed Data." Ann. Appl. Probab. 2 (1) 142 - 170, February, 1992.


Published: February, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0742.60026
MathSciNet: MR1143397
Digital Object Identifier: 10.1214/aoap/1177005775

Primary: 60F10
Secondary: 62F99

Keywords: Gibbs fields , large deviations , maximum likelihood , Variational principle

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 1 • February, 1992
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