Inverse Of Fisher Information Matrix Research Articles

An Interpretation of the Moore-Penrose Generalized Inverse of a Singular Fisher Information Matrix

It is proved that in a non-Bayesian parametric estimation problem, if the Fisher information matrix (FIM) is singular, unbiased estimators for the unknown parameter will not exist. Cramer-Rao bound (CRB), a popular tool to lower bound the variances of unbiased estimators, seems inapplicable in such situations. In this paper, we show that the Moore-Penrose generalized inverse of a singular FIM can be interpreted as the CRB corresponding to the minimum variance among all choices of minimum constraint functions. This result ensures the logical validity of applying the Moore-Penrose generalized inverse of an FIM as the covariance lower bound when the FIM is singular. Furthermore, the result can be applied as a performance bound on the joint design of constraint functions and unbiased estimators.

Quantum measurement and information

AbstractThe operationally defined invariant information introduced by Brukner and Zeilinger is related to the problem of estimation of quantum states. It quantifies how the estimated states differ in average from the true states in the sense of Hilbert‐Schmidt norm. This information evaluates the quality of the measurement and data treatment adopted. Its ultimate limitation is given by the trace of inverse of Fisher information matrix.

Optimal delay estimation in a multiple sensor array having spatially correlated noise

The maximal likelihood (ML) estimation of time-of-arrival differences for signals from a single source or target arriving at M \geq 2 sensors has been the subject of a large number of papers in recent years. These time differences or delays enable target location. Nearly all previous work has assumed noises which are independent among all sensors. Herein, noises are taken to have complex correlation between sensors. A set of nonlinear equations in the unknown delays is derived and the Fisher information matrix (FIM) for the estimates is also derived. The Cramer-Rao matrix bound (CRMB), which is the inverse of FIM, shows optimal estimator covariances. Computer evaluations are plotted for CRMB elements with varied SNR and noise covariance values typical of turbulent boundary layer noise in towed arrays and signal sources at infinite range (plane-wave fronts). Maximum changes in the bound are within ±3 dB for complex noise correlations with magnitudes up to 0.4, which we tested.