  The Fisher information matrix (FIM) is a foundational concept in statistical
signal processing. The FIM depends on the probability distribution, assumed to
belong to a smooth parametric family. Traditional approaches to estimating the
FIM require estimating the probability distribution function (PDF), or its
parameters, along with its gradient or Hessian. However, in many practical
situations the PDF of the data is not known but the statistician has access to
an observation sample for any parameter value. Here we propose a method of
estimating the FIM directly from sampled data that does not require knowledge
of the underlying PDF. The method is based on non-parametric estimation of an
$f$-divergence over a local neighborhood of the parameter space and a relation
between curvature of the $f$-divergence and the FIM. Thus we obtain an
empirical estimator of the FIM that does not require density estimation and is
asymptotically consistent. We empirically evaluate the validity of our approach
using two experiments.
