Use and Subtleties of Saddlepoint Approximation for Minimum Mean-Square Error Estimation

An integral representation for the minimum mean-square error (MMSE) estimator for a random variable in an observation model consisting of a linear combination of two random variables is derived. The derivation is based on the moment-generating functions for the random variables in the observation model. The method generalizes so that integral representations for higher-order moments of the posterior of interest can be easily obtained. Two examples are presented that demonstrate how saddle-point approximation can be used to obtain accurate approximations for a MMSE estimator using the derived integral representation. However, the examples also demonstrate that when two saddle points are close or coalesce, then saddle-point approximation based on isolated saddle points is not valid. A saddle-point approximation based on two close or coalesced saddle points is derived and in the examples, the validity and accuracy of the derivation is demonstrated.