The joint MAP-ML criterion and its relation to ML and to extended least-squares

The joint maximum a posteriori-maximum likelihood (JMAP-ML) estimation criterion can serve as an alternative to the maximum likelihood (ML) criterion when estimating parameters from an observed data vector whenever another unobserved data vector is involved. Rather than maximize the probability of the observed data with respect to the parameters, JMAP-ML maximizes the joint probability of the observed and unobserved data with respect to both the unknown parameters and the unobserved data. In this paper, we characterize the relation between the ML and JMAP-ML estimates in the Gaussian case and provide insight into the apparent bias of JMAP-ML. Although JMAP-ML is an inconsistent estimator, we show that with short data records, it is often preferable to ML in terms of both bias and variance. We also identify JMAP-ML as a special case of the deterministic extended least squares (XLS) criterion. We indicate a general relation between a possible maximization algorithm for JMAP-ML and the well-known estimation-maximization (EM) algorithm.