Estimation of a Noisy Discrete-Time Step Function: Bayes and Empirical Bayes Approaches

Consider the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function. Best linear estimates and Bayes estimates are derived, evaluated and compared. A characterization of the Bayes estimates is presented. This characterization has an intuitive interpretation and also provides a way to compute the Bayes estimates with a number of operations of the order of T³ where T is the fixed time span. An approximation to the Bayes estimates is proposed which reduces the total number of operations to the order of T. The results are applied to the case where the Bayesian model fails to be satisfied using an empirical Bayes approach.