We introduce the concept of a propagator function to characterize unilateral Markov fields that fulfil the positivity condition. We show that parameter estimation of a model realization corresponds to the solution of a given set of equations, which can be solved explicitly for a specific propagator function depending exclusively on identities among site labels. For a hidden model, we propose a nonsupervised, iterative scheme comprising labeling and parameter re-estimation steps. Segmentation experiments using a third order Markov mesh, together with a look-ahead algorithm for real-time label estimation, show rapid convergence of the learning algorithm and yield subjectively good results.

The image modeling problem is discussed under the assumption that images can be represented by third-order. hidden Markov mesh random field models. The modeling applications comprise restoration of binary images, compression of image data, and segmentation of gray-level images and image sequences under the short-range motion hypothesis. Coherent approaches to the problems of image modeling and estimation of model parameters are outlined. A labeling algorithm based on a maximum marginal a posteriori probability criterion is proposed. Critical aspects of the computer simulation of a real-time implementation are discussed in detail. A learning technique by which the model parameters can be estimated without ground truth information is developed. Extensive experimentation with both static and dynamic images from a variety of sources is discussed.>

The problem of estimating the error probability of a given classification system is considered. Statistical properties of the empirical error count (C) and the average conditional error (R) estimators are studied. It is shown that in the large sample case the R estimator is unbiased and its variance is less than that of the C estimator. In contrast to conventional methods of Bayes error estimation the unbiasedness of the R estimator for a given classifier can be obtained only at the price of an additional set of classified samples. On small test sets the R estimator may be subject to a pessimistic bias caused by the averaging phenomenon characterizing the functioning of conditional error estimators.

Observations are made relative to the definition and generation of a consistent training sample set. We show that the consistency assumption is crucial to the method described in the above correspondence,1 and that, in general, the proposed solution to the generation problem, namely k-NN editing, fails to produce consistent sets.

The probability distribution of the error incurred by a classification system on a given data set is derived. It is shown that the distribution is mixed binomial. A method for calculating the mixed binomial distribution recursively is proposed.

A distribution-free lower bound on the Bayes error rate is formulated in terms of the asymptotic error rate of the nearest neighbor rule with a reject option. Next, a closed form expression for an upper bound of thekth nearest neighbor error rate in terms of the Bayes rate is established. These results are discussed in the framework of recent works on nonparametric estimation of the Bayes error rate.

An important measure concerning the use of statistical decision schemes is the error probability associated with the decision rule. Several methods giving bounds on the error probability are presently available, but, most often, the bounds are loose. Those methods generally make use of so-cailed distances between statistical distributions. In this paper a new distance is proposed which permits tighter bounds to be set on the error probability of the Bayesian decision rule and which is shown to be closely related to several certainty or separability measures. Among these are the nearest neighbor error rate and the average conditional quadratic entropy of Vajda. Moreover, our distance bears much resemblance to the information theoretic concept of equivocation. This relationship is discussed. Comparison is made between the bounds on the Bayes risk obtained with the Bhattacharyya coefficient, the equivocation, and the new measure which we have named the Bayesian distance.