  A new nonparametric approach for system identification has been recently
proposed where the impulse response is seen as the realization of a zero--mean
Gaussian process whose covariance, the so--called stable spline kernel,
guarantees that the impulse response is almost surely stable. Maximum entropy
properties of the stable spline kernel have been pointed out in the literature.
In this paper we provide an independent proof that relies on the theory of
matrix extension problems in the graphical model literature and leads to a
closed form expression for the inverse of the first order stable spline kernel
as well as to a new factorization in the form $UWU^\top$ with $U$ upper
triangular and $W$ diagonal. Interestingly, all first--order stable spline
kernels share the same factor $U$ and $W$ admits a closed form representation
in terms of the kernel hyperparameter, making the factorization computationally
inexpensive. Maximum likelihood properties of the stable spline kernel are also
highlighted. These results can be applied both to improve the stability and to
reduce the computational complexity associated with the computation of stable
spline estimators.
