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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?

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  • $\begingroup$ Whether or not a second order model is better than a first order model in terms of state estimation ultimately depends on the actual (real-world) system you are modelling. If the speed is constant in reality, a second order model might do harm to your estimation whereas a first order model would be inappropriate for an application in which the speed is not constant. This has nothing to do with Kalman filter or any other tool you may want to use. $\endgroup$ – Calculon Apr 27 '18 at 7:42
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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).

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