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

Abstract

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 $T^3$ 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.