If we're trying to track the position of an object and we apply a Kalman filter in order to estimate its location, how can we assess how good the estimates are if we do not know the true positions? Let's say we only have access to some coordinates, which are the true positions plus/minus some Gaussian error.
If I have two models for position e.g, a first order and a second order approximation, can you actually assess which model provides better estimates?
This is not a definitive answer but if you have access to true positions w/ Gaussian noise, you may want to minimize the negative log likelihood:
$\mathcal{L}=-\sum_i \log\left(p_{\text{Kalman}}(x_i)\right)$
where $x_i$ are your measurements and $p_{\text{Kalman}}$ is the estimate of the probability distribution by the Kalman filter (i.e. the mean and the variance).
I hope that helps (I'm sorry I am not allowed to comment on the question so I've to put this in the answer field).