In Kalman filter, what is the diagnosis when the variance-covariance matrix of the updated distribution is progressively increasing? Let $\mathbf{\theta}_t$ be a state vector at time $t$ and $p(\mathbf{\theta}_t | \mathbf{y}_{1:t}) = \mathrm{N}(\mathbf{m}_t, \mathbf{C}_t)$ be its posterior distribution. What can I say of the model if $\mathbf{C}_t$ is increasing when $t \to \infty $. My current hypothesis is that the data do not give enough information to estimate the correct values of $\mathbf{\theta}_t$.
 A: If you think about the state vector of a point, just x and y, you could plot the covariance matrix as an ellipse. The covariance matrix is just the uncertainty of the state vector at time t, so if it increases, it means that the system is every time more uncertain. About the reasons, perhaps the data that you are giving to the system does not make any sense? And therefore, you are confusing the system rather than giving it good data.
If you think about a ship, for instance. If in the first iteration it takes a direction, it goes to the front, and the observation agrees, then the probability of the ship in the next iteration to go to the front is much higher, the covariance matrix in that direction decreases, because it is more certain that it will go in that direction.
If in the next iteration the same happens, the variance in that direction will keep decreasing.
Kalman Filter is able to work without all the information, because it has the prior that kind of speaks for the missing information, however, I am not sure about what happens to the covariance of the features when an observation is missing. My guess is that it increases, given that no new data has been provided. If so, then your hypothesis would be another potential reason for the covariance to increase.
