  In this paper we provide novel closed-form expressions enabling
differentiation of any scalar function of the Kalman filter's outputs with
respect to all its tuning parameters and to the measurements. The approach
differs from the previous well-known sensitivity equations in that it is based
on a backward (matrix) gradient calculation, that leads to drastic reductions
of the overall computational cost. It is our hope that practitioners seeking
numerical efficiency and reliability will benefit from the concise and exact
equations derived in this paper and the methods that build upon them. They may
notably lead to speed-ups when interfacing a neural network with a Kalman
filter.
