Likelihood ratios for sequential hypothesis testing on Markov sequences

A variety of likelihood ratios are derived for detecting Gauss-Markov and finite-state Markov sequences in additive Gaussian noise. The Bayesian recursions appropriate to related filtering problems are exploited, together with "known-form" likelihood ratios, to obtain the desired results. In the derivation of a discrete-time Gauss-Markov likelihood ratio, a "pure" causal estimator-correlator structure is sought and a "locally stable" state estimator is encountered that is of some interest in its own right. The likelihood ratio is "pure" in the sense that the locally stable estimator is used in precisely the same manner as the stored replica is used in known-form signal detection problems to form the likelihood ratio. Consequently, the likelihood ratio is devoid of the extra data-dependent term that arises whenever one uses least squares state estimators to form the likelihood ratio statistic. The locally stable estimator equalizes, within a constant related to the {em a priori} and {em a posteriori} filtering error covariances, the {em a priori} and {em a posteriori} filtering densities. Heuristically, the estimator is a compromise between the one-step predictor and the filtered estimator of a discrete-time Kalman filter. When the observation noise covariance is unknown, a generalization of the so-called unknown level problem, then a Wishart prior is assigned to the innovations covariance and an integral representation is obtained for the desired likelihood ratio. The representation suggests a parallel structure for approximating the likelihood ratio when the observation noise covariance is unknown. Finally, the likelihood ratio for detecting finite-state Markov sequences is derived to illustrate that in general no "pure" estimator-correlator structure can exist when the state-space is finite.