The Annals of Applied Probability

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

Murilo P. Almeida and Basilis Gidas

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Abstract

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 $\mathbb{R}^n, n \geq 1$, 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.

Article information

Source
Ann. Appl. Probab., Volume 3, Number 1 (1993), 103-136.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005510

Digital Object Identifier
doi:10.1214/aoap/1177005510

Mathematical Reviews number (MathSciNet)
MR1202518

Zentralblatt MATH identifier
0771.60091

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J99: None of the above, but in this section 62H12: Estimation 62M40: Random fields; image analysis

Keywords
Super-stable Gibbs distributions variational estimators consistency asymptotic normality

Citation

Almeida, Murilo P.; Gidas, Basilis. A Variational Method for Estimating the Parameters of MRF from Complete or Incomplete Data. Ann. Appl. Probab. 3 (1993), no. 1, 103--136. doi:10.1214/aoap/1177005510. https://projecteuclid.org/euclid.aoap/1177005510


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