Fisher Information Bounds with Applications in Nonlinear Learning, Compression and Inference

The problem how to derive a generic lower bound for the Fisher information measure is considered. We review a recent approach by two examples and identify a connection between the construction of strong Fisher information bounds and the sufficient statistics of the underlying system model. In order to present the problem of such information bounds within a broad scope, we discuss the properties of the Fisher information measure for distributions belonging to the exponential family. Under this restriction, we establish an identity connecting Fisher information, the natural parameters and the sufficient statistics of the system model. Replacing an arbitrary system model by an equivalent distribution within the exponential family, we then derive a general lower bound for the Fisher information measure. With the optimum estimation theoretic model matching rule we show how to obtain a strong version of the information bound. We then demonstrate different applications of the proposed conservative likelihood framework and the derived Fisher information bound. In particular, we discuss how to determine the minimum guaranteed inference capability of a memoryless system with unknown statistical output model and show how to achieve this pessimistic performance assessment with a root-n consistent estimator operating on a nonlinear compressed version of the observed data. Finally, we identify that the derived conservative maximum-likelihood algorithm can be formulated as a special version of Hansen’s generalized method of moments.