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Register a new userInformation Geometry and Classical Cram{e}r-Rao Type Inequalities
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Added by
Kumar Vijay Mishra
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We examine the role of information geometry in the context of classical Cramer-Rao (CR) type inequalities. In particular, we focus on Eguchis theory of obtaining dualistic geometric structures from a divergence function and then applying Amari-Nagoakas theory to obtain a CR type inequality. The classical deterministic CR inequality is derived from Kullback-Leibler (KL)-divergence. We show that this framework could be generalized to other CR type inequalities through four examples: $alpha$-version of CR inequality, generalized CR inequality, Bayesian CR inequality, and Bayesian $alpha$-CR inequality. These are obtained from, respectively, $I_alpha$-divergence (or relative $alpha$-entropy), generalized Csiszar divergence, Bayesian KL divergence, and Bayesian $I_alpha$-divergence.
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The variance and the entropy power of a continuous random variable are bounded from below by the reciprocal of its Fisher information through the Cram{e}r-Rao bound and the Stams inequality respectively. In this note, we introduce the Fisher information for discrete random variables and derive the discrete Cram{e}r-Rao-type bound and the discrete Stams inequality.
It is challenged only recently that the precision attainable in any measurement of a physical parameter is fundamentally limited by the quantum Cram{e}r-Rao Bound (QCRB). Here, targeting at measuring parameters in strongly dissipative systems, we propose an innovative measurement scheme called {it dissipative adiabatic measurement} and theoretically show that it can beat the QCRB. Unlike projective measurements, our measurement scheme, though consuming more time, does not collapse the measured state and, more importantly, yields the expectation value of an observable as its measurement outcome, which is directly connected to the parameter of interest. Such a direct connection {allows to extract} the value of the parameter from the measurement outcomes in a straightforward manner, with no fundamental limitation on precision in principle. Our findings not only provide a marked insight into quantum metrology but also are highly useful in dissipative quantum information processing.
In this paper, we analyze the impact of compressed sensing with complex random matrices on Fisher information and the Cram{e}r-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-orthogonally invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.
Single molecule localization microscopy has the potential to resolve structural details of biological samples at the nanometer length scale. However, to fully exploit the resolution it is crucial to account for the anisotropic emission characteristics of fluorescence dipole emitters. In case of slight residual defocus, localization estimates may well be biased by tens of nanometers. We show here that astigmatic imaging in combination with information about the dipole orientation allows to extract the position of the dipole emitters without localization bias and down to a precision of ~1nm, thereby reaching the corresponding Cram{e}r Rao bound. The approach is showcased with simulated data for various dipole orientations, and parameter settings realistic for real life experiments.
In this article, we describe various aspects of categorification of the structures appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F-manifolds, and motivic enrichments.
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