Cramer Rao maximum a-posteriori bounds for a finite number of non-Gaussian parameters

In minimum mean square estimation, an estimate theta' of the M-dimensional random parameter vector theta is obtained from a noisy N-dimensional input vector y. In this paper, we develop CRM bounds for the case where y and theta are non-Gaussian and M is small. First, y is linearly transformed to x(theta) which is approximately Gaussian because of the central limit theorem (CLT). Second, an arbitrary auxiliary signal model is introduced with Gaussian input vector x(d) and Gaussian parameter vector d which are statistically independent of y and theta. Then an augmented signal model is formed with augmented input vector x(a) = [x(theta)\x(d)](T) and augmented parameter vector theta(a) = [theta\d](T). The CRM bounds for phi(a) are then transformed into CRM bounds for theta(a). Consequently, CRM bounds on theta can be calculated from the signal model x(theta) as if theta were Gaussian.