Cramer-Rao Bound Analog of Bayes' Rule [Lecture Notes]

The estimation of multiple parameters is a common task in signal processing. The Cramer-Rao bound (CRB) sets a statistical lower limit on the resulting errors when estimating parameters from a set of random observations. It can be understood as a fundamental measure of parameter uncertainty [1], [2]. As a general example, suppose denotes the vector of sought parameters and that the random observation model can be written as y = xi + w, (1) where xi is a function or signal parameterized by i and w is a zero-mean Gaussian noise vector. Then the CRB for i has the following notable properties: 1) For a fixed i, the CRB for i decreases as the dimension of y increases. 2) For a fixed y, if additional parameters i u are estimated, then the CRB for i increases as the dimension of i u increases. 3) If adding a set of observations yu requires estimating additional parameters, i u then the CRB for i decreases as the dimension of yu increases, provided the dimension of i u does not exceed that of yu [3]. This property implies both 1) and 2) above. 4) Among all possible distributions of w with a fixed covariance matrix, the CRB for i attains its maximum when w is Gaussian, i.e., the Gaussian scenario is the "worst case" for estimating θ [4]-[6].