A method for approximating the density of maximum-likelihood and maximum a posteriori estimates under a Gaussian noise model

The performance of maximum-likelihood (ML) and maximum a posteriori (MAP) estimates in non-linear problems at low data SNR is not well predicted by the Cramér-Rao or other lower bounds on variance. In order to better characterize the distribution of ML and MAP estimates under these conditions, we derive a point approximation to density values of the conditional distribution of such estimates. In an example problem, this approximate distribution captures the essential features of the distribution of ML estimates in the presence of Gaussian-distributed noise.