# The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively. $$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$ $$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$

In the case of Gaussian process and measurement noises, suppose $$P_k$$ is the posterior estimation error covariance matrix of the Kalman filter, $$\tilde{P}_k$$ is the estimation error covariance matrix of an arbitrary estimator (maybe nonlinear). Since Kalman filter is an MMSE estimator, we know

$$Trace(\tilde{P}_k-P_k) \geq 0$$

Now I wonder whether a more strong conclusion is true, i.e.,

$$\tilde{P}_k-P_k \succeq 0$$

which means that $$\tilde{P}_k-P_k$$ is always positive semi-definite. For an arbitrary linear filter, I know this is true; but I am not sure for an arbitrary nonlinear estimator. Any reference paper is appreciated.

• Do you want the estimator to be unbiased? Jun 4 at 7:23
• @S.Surace No, an arbitrary is not necessarily unbiased. In this case, \tilde{P}_k denotes the second moment of estimation error. Jun 7 at 3:34