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Winkler, Gerhard (1998): An Adaptive Gradient Algorithm for Maximum Likelihood Estimation in Imaging. A Tutorial. Collaborative Research Center 386, Discussion Paper 120




Markov random fields serve as natural models for patterns or textures with random fluctuations at small scale. Given a general form of such fields each class of pattern corresponds to a collection of model parameters which critically determines the abilitity of algorithms to segment or classify. Statistical inference on parameters is based on (dependent) data given by a portion of patterns inside some observation window. Unfortunately, the corresponding maximum likelihood estimators are computationally intractable by classical methods. Until recently, they even were regarded as intractable at all. In recent years stochastic gradient algorithms for their computation were proposed and studied. An attractive class of such algorithms are those derived from adaptive algorithms, wellknown in engeneering for a long time. We derive convergence theorems following closely the lines proposed by M. Metivier and P. Priouret (1987). This allows a transparent (albeit somewhat technical) treatment. The results are weaker than those obtained by L. Younes (1988). Keywords: adaptive algorithm, stochastic approximation, stochastic gradient descent, MCMC methods, maximum likelihood, Gibbs fields, imaging