# Why the Kalman filter doesn't increase its dispersion when its predictions are wrong?

Assume for simplicity that the problem is 1d, the transition we are studying is the very simple:

$x_{t+1} = x_t + \epsilon$
$z_{t} = x_t + u$

The measurement-update phase of the 1d Kalman filter looks like this:
$k = \frac{p_{t-1}}{p_{t-1}+\sigma_u^2}$
$\hat x_{t} = \hat x_{t-1} + k_d (z_t - \hat x_{t-1})$
$p_t = p_{t-1} - p_{t-1} k$

What I don't understand is why $p$ (the estimated error variance) decreases at every measurement regardless of how bad the prediction is. It seems to me that when the prediction is off by a lot we should increase our uncertainty on $\hat x$ rather than decrease it.

• An alternative way to write the last equation seems to be $p_t=(1-k)p_{t-1}=\frac{p_{t-1}\sigma^2_u}{p_{t-1}+\sigma^2_u}$, maybe that can help to see something (I don't know). Aug 10 '16 at 16:15