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Empirical non-parametric estimation of the Fisher Information

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.