The Annals of Applied Probability

Parameter Estimation for Gibbs Distributions from Partially Observed Data

Francis Comets and Basilis Gidas

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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.

Article information

Ann. Appl. Probab., Volume 2, Number 1 (1992), 142-170.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F10: Large deviations
Secondary: 62F99: None of the above, but in this section

Gibbs fields large deviations maximum likelihood variational principle


Comets, Francis; Gidas, Basilis. Parameter Estimation for Gibbs Distributions from Partially Observed Data. Ann. Appl. Probab. 2 (1992), no. 1, 142--170. doi:10.1214/aoap/1177005775.

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