I came a cross a property for Kalman filter known as invariant property. I could only find some information about it on a wikipedia article but I still struggle to understand it.
The property is that if $E[x_k-\hat{x}_k]=0$, where $x_k$ is the state vector at time $k$ and $\hat{x}_k$ is the estimate of $x_k$, then $$COV(x_k-\hat{x}_k)=COV(\hat{x}_k) \tag1\label{1}$$, where $COV(x_k-\hat{x}_k)$ corresponds to the updated estimate covariance.
I am puzzled by this because on one hand, \eqref{1} implies that $$trace(COV(x_k-\hat{x}_k))=trace(COV(\hat{x}_k))$$ and on other other hand, $trace(COV(x_k-\hat{x}_k))=MSE(\hat{x}_k)$, and hence$$MSE(\hat{x}_k)=trace(COV(\hat{x}_k))+bias\tag2\label2$$ and the $bias$ is some function of $E[x_k-\hat{x}_k]$ and it is zero if and only if the invariant holds. So my questions are:
- Is \eqref{2} correct? If it is not then what is the implication of invariant property for $MSE(\hat{x}_k)$?
- If \eqref{2} is correct, then is it minimized by setting the bias to zero?
- If Kalman for linear Gaussian system is optimal in terms of minimizing MSE, doesn't this make Kalman filtering and unbiased estimator? (how could a Bayesian estimator be unbiased?)
- Finally, if all the above are based on my wrong understanding of invariant property, could someone elaborate on it. In particular, I am interested to know its implications on MSE, for example could I say a Kalman with invariant is optimum in MSE sense and outperforms Kalman without invriant propert?