A Bayesian decision theory approach is applied to the solution of the problem of unsupervised parametric pattern recognition. The parametric model for this investigation includes the cases where both constant and time-varying unknown parameters are present, and, most significantly, the unknown hypotheses do not constitute a statistically independent sequence. They are restricted only to be from a source with finite-order Markov dependence. The resulting optimal learning system is found and shown to grow initially in size and memory until theNth observation (whereNis the highest Markov order), and subsequently to remain of fixed size and memory. It can, therefore, operate indefinitely and continue to improve its ability to recognize patterns utilizing only a fixed size memory. In summary, the main contributions of this paper are the following: \begin{enumerate} \item the extension of previous investigations of the unsupervised parametric pattern recognition problem to include cases where both constant and time-varying unknown parameter vectors are simultaneously present; \item that the a priori probabilities of the hypotheses, the time-varying parameters, and their transition laws may, if constant, be expressed as functions of the constant unknown parameter and, thus, also be learned; and \item the removal of the assumption of statistical independence between hypotheses for the sequence of observations. \end{enumerate}

Recursive relations are given for updating the conditional densityp(\theta_{k} | X_{k-1}, \cdots X_{1})(also forp(\theta_{k} | X_{k}, \cdots, X_{1})), where\theta_{k}is a parameter of the density ofX_{k}. The observationsX_{1}, X_{2}, \cdotsare assumed to be conditionally independent (i.e., for known parameters), and the sequence of time-varying parameters\theta_{1}, \theta_{2}, \cdotsconstitutes a Markov-M sequence. The result requires the storage of an intermediate function of(\theta_{k-1}, \cdots , \theta_{k-M}).