Cramér-Rao lower bound for linear filtering with t-distributed measurement noise

The Cramér-Rao lower bound (CRLB) on the achievable mean square error (MSE) can be used to evaluate approximate estimation algorithms. For linear filtering problems with non-Gaussian noises, the CRLB can be easily computed using the Kalman filter state covariance recursion with the Fisher information in place of the noise covariance term. This work studies a linear filtering problem with t-distributed measurement noise. It is found that for a t distribution with heavy tails, the CRLB significantly underestimates the optimal MSE, the Kalman filter has significantly larger MSE, and a computationally light variational-Bayes algorithm achieves nearly optimal MSE.