A Variational Method for Estimating the Parameters of MRF from Complete or Incomplete Data

We introduce a new method (to be referred to as the variational method, VM) for estimating the parameters of Gibbs distributions with random variables ("spins") taking values in a Euclidean space Rn, n ≥ 1, from complete or degraded data. The method applies also to the case of iid random v...

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Bibliographic Details
Published in:The Annals of applied probability 1993, Vol.3 (1), p.103-136
Main Authors: Almeida, Murilo P., Gidas, Basilis
Format: Article
Language:English
Subjects:
60J99
60K35
62H12
62M40
Algorithms
asymptotic normality
consistency
Covariance
Ergodic theory
Estimate reliability
Estimation methods
Estimators
Maximum likelihood estimation
Pixels
Random variables
Super-stable Gibbs distributions
variational estimators
Variational methods
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Summary:We introduce a new method (to be referred to as the variational method, VM) for estimating the parameters of Gibbs distributions with random variables ("spins") taking values in a Euclidean space Rn, n ≥ 1, from complete or degraded data. The method applies also to the case of iid random variables governed by exponential families, and appears to be new even in this case. For complete data, the VM is computationally more efficient than, and as reliable as, the maximum pseudo-likelihood method. For incomplete data, the VM leads to an estimation procedure reminiscent of, but simpler than, the EM algorithm. In the former case, we show that under natural assumptions a certain form of the variational estimators is strongly consistent and asymptotically normal. We also present numerical experiments that demonstrate the computational efficiency and accuracy of the variational estimators.
ISSN:1050-5164
2168-8737
DOI:10.1214/aoap/1177005510