  In this paper, we propose an approach to address the problems with ambiguity
in tuning the process and observation noises for a discrete-time linear Kalman
filter. Conventional approaches to tuning (e.g. using normalized estimation
error squared and covariance minimization) compute empirical measures of filter
performance and the parameter are selected manually or selected using some kind
of optimization algorithm to maximize these measures of performance. However,
there are two challenges with this approach. First, in theory, many of these
measures do not guarantee a unique solution due to observability issues.
Second, in practice, empirically computed statistical quantities can be very
noisy due to a finite number of samples. We propose a method to overcome these
limitations. Our method has two main parts to it. The first is to ensure that
the tuning problem has a single unique solution. We achieve this by
simultaneously tuning the filter over multiple different prediction intervals.
Although this yields a unique solution, practical issues (such as sampling
noise) mean that it cannot be directly applied. Therefore, we use Bayesian
Optimization. This technique handles noisy data and the local minima that it
introduces.
