The Kalman recursion for state space models is extended to allow for likelihood evaluation and minimum mean square estimation given states with an arbitrarily large covariance matrix. The extension is computationally minor. Application is made to likelihood evaluation, state estimation, prediction and smoothing.
"The Diffuse Kalman Filter." Ann. Statist. 19 (2) 1073 - 1083, June, 1991. https://doi.org/10.1214/aos/1176348139