An empirical Bayes decision problem

The empirical Bayes decision problem is considered. Let {ai}, i=1,..,N be a sequence of Markov dependent random variables, ai∈ {1,...,m} where ai denotes the category of the ith sample also called the state of nature. Let pl j be the elements of the transition matrix of the Markov process and consider that the initial probabilities are equal to the steady state probabilities of the Markov chain. Let xN = {x1,...,xN} be a sequence of random observations where xi has probability density function fa i(xi). Suppose that the receiver does not know which state of nature is acting after the reception of the sample xi and after N observations, it is desired to partition the received samples into m sets with minimum probability of misclassification with respect to the true partition induced by the states of nature. Such a problem may arise in recognition of written characters Ref. [4] and in receiving signals over a noisy channel with intersymbol interference Ref. [3]. In the present work it is assumed that the fj(xi) are unknown, it is only known that fj(xi) belong to a family F of pdf's with known functional form. It is assumed that the probability transition matrix of the Markov chain is unknown. It is shown that if the family F of pdf's satisfies certain identifiability and differentiality conditions, then by using moment estimates of the unknown quantities, a decision function t can be determined such that the corresponding risk converges to the optimal Bayes risk. The present work extends the results obtained in Ref. [4] by considering that the transition probabilities and the pdf's are unknown. The work of Ref. [3] is extended by showing file convergence of the risk corresponding to t to the optimal risk, without requiring that the signal to noise ratio converge to zero.