Approximate distributions for 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 nonlinear problems at low data SNR is not well predicted by the Cramer-Rao or other lower bounds on variance. In order to better characterize the distribution of ML and MAP estimates under these conditions, we derive an approximate density for the conditional distribution of such estimates. In one example, this approximate distribution captures the essential features of the distribution of ML and MAP estimates in the presence of Gaussian-distributed noise.