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A general theory for the tracking and classification of multiple targets using a Bayesian approach is presented, together with its specialization to independent, identically distributed target models. The implementation of the theory is through the Generalized Tracker/Classifier. Simulation results to illustrate the algorithm are also given.

Bayesian field theory denotes a nonparametric Bayesian approach for learning functions from observational data. Based on the principles of Bayesian statistics, a particular Bayesian field theory is defined by combining two models: a likelihood model, providing a probabilistic description of the measurement process, and a prior model, providing the information necessary to generalize from training to non-training data. The particular likelihood models discussed in the paper are those of general density estimation, Gaussian regression, clustering, classification, and models specific for inverse quantum problems. Besides problem typical hard constraints, like normalization and positivity for probabilities, prior models have to implement all the specific, and often vague, "a priori" knowledge available for a specific task. Nonparametric prior models discussed in the paper are Gaussian processes, mixtures of Gaussian processes, and non-quadratic potentials. Prior models are made flexible by including hyperparameters. In particular, the adaption of mean functions and covariance operators of Gaussian process components is discussed in detail. Even if constructed using Gaussian process building blocks, Bayesian field theories are typically non-Gaussian and have thus to be solved numerically. According to increasing computational resources the class of non-Gaussian Bayesian field theories of practical interest which are numerically feasible is steadily growing. Models which turn out to be computationally too demanding can serve as starting point to construct easier to solve parametric approaches, using for example variational techniques.

The task of inferring a set of classes and class descriptions most likely to explain a given data set can be placed on a firm theoretical foundation using Bayesian statistics. Within this framework and using various mathematical and algorithmic approximations, the AutoClass system searches for the most probable classifications, automatically choosing the number of classes and complexity of class descriptions. A simpler version of AutoClass has been applied to many large real data sets, has discovered new independently-verified phenomena, and has been released as a robust software package. Recent extensions allow attributes to be selectively correlated within particular classes, and allow classes to inherit or share model parameters though a class hierarchy. We summarize the mathematical foundations of AutoClass.