  This paper considers the problem of fitting the parameters of a Kalman
smoother to data. We formulate the Kalman smoothing problem with missing
measurements as a constrained least squares problem and provide an efficient
method to solve it based on sparse linear algebra. We then introduce the Kalman
smoother tuning problem, which seeks to find parameters that achieve low
prediction error on held out measurements. We derive a Kalman smoother
auto-tuning algorithm, which is based on the proximal gradient method, that
finds good, if not the best, parameters for a given dataset. Central to our
method is the computation of the gradient of the prediction error with respect
to the parameters of the Kalman smoother; we describe how to compute this at
little to no additional cost. We demonstrate the method on population migration
within the United States as well as data collected from an IMU+GPS system while
driving. The paper is accompanied by an open-source implementation.
