I'm working through the derivation of the Kalman filter equations from this paper (or alternative source here) and I'm unsure of the derivation of the state prediction covariance (equation 2 in the paper).
I understand that we want $\mathbf{P}_{t|t-1}$ to be the covariance of the error of the prediction from the true value, i.e. $\text{Cov}\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)$, but the author proceeds as follows:
$\mathbf{P}_{t|t-1} = \text{E}\left[\text{Cov}\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)\right] = \text{E}\left[\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)^\top\right]$
My question is, why do we need the expectation? i.e., why is this not just
$\mathbf{P}_{t|t-1} = \text{Cov}\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right) = \left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)\left(\mathbf{x}_t-\hat{\mathbf{x}}_{t|t-1}\right)^\top$
This would seem to work through the derivation in the same way to give the same end result.
Put another way, what's the expectation of a covariance?