Filtering and prediction from uncertain observations with correlated signal and noise via mixture approximations

A recursive suboptimal filtering and prediction algorithm is designed to estimate Gaussian signals from uncertain observations, when the variables describing the uncertainty are independent, and the signal and observation white noise are correlated. The derivation is based on successive approximations of mixtures of normal distributions which, in turn, provide approximations of the conditional distributions of the signal given the observations. The proposed estimators, which are nonlinear functions of the observations, are then obtained as the expectation of these approximate conditional distributions. The algorithm does not require the whole knowledge of the state-space model generating the signal, but only covariance information of the signal and the observation noise, as well as the probability that the signal exists in the observed values.