Fisher information metric

Summary

In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements. The metric is interesting in several respects. By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. It can also be understood to be the infinitesimal form of the relative entropy (i.e., the Kullback–Leibler divergence); specifically, it is the Hessian of the divergence. Alternately, it can be understood as the metric induced by the flat space Euclidean metric, after appropriate changes of variable. When extended to complex projective Hilbert space, it becomes the Fubini–Study metric; when written in terms of mixed states, it is the quantum Bures met

Official source

https://en.wikipedia.org/wiki/Fisher_information_metric

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We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas and techniques that come from probability, information theory as well as signal processing.

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Differential Entropy of the Conditional Expectation under Gaussian Noise

Ahmet Arda Atalik, Michael Christoph Gastpar, Alper Köse

This paper considers an additive Gaussian noise channel with arbitrarily distributed finite variance input signals. It studies the differential entropy of the minimum mean-square error (MMSE) estimator and provides a new lower bound which connects the differential entropy of the input, output, and conditional mean. That is, the sum of differential entropies of the conditional mean and output is always greater than or equal to twice the input differential entropy. Various other properties such as upper bounds, asymptotics, Taylor series expansion, and connection to Fisher Information are obtained. An application of the lower bound in the remote-source coding problem is discussed, and extensions of the lower and upper bounds to the vector Gaussian channel are given.

IEEE2021

Differential Entropy of the Conditional Expectation Under Additive Gaussian Noise

Ahmet Arda Atalik, Michael Christoph Gastpar, Alper Köse

The conditional mean is a fundamental and important quantity whose applications include the theories of estimation and rate-distortion. It is also notoriously difficult to work with. This paper establishes novel bounds on the differential entropy of the conditional mean in the case of finite-variance input signals and additive Gaussian noise. The main result is a new lower bound in terms of the differential entropies of the input signal and the noisy observation. The main results are also extended to the vector Gaussian channel and to the natural exponential family. Various other properties such as upper bounds, asymptotics, Taylor series expansion, and connection to Fisher Information are obtained. Two applications of the lower bound in the remote-source coding and CEO problem are discussed.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Strengthened Information-theoretic Bounds on the Generalization Error

Amedeo Roberto Esposito, Michael Christoph Gastpar, Ibrahim Issa

The following problem is considered: given a joint distribution P XY and an event E, bound P XY (E) in terms of P X P Y (E) (where P X P Y is the product of the marginals of P XY ) and a measure of dependence of X and Y. Such bounds have direct applications in the analysis of the generalization error of learning algorithms, where E represents a large error event and the measure of dependence controls the degree of overfitting. Herein, bounds are demonstrated using several information-theoretic metrics, in particular: mutual information, lautum information, maximal leakage, and J ∞ . The mutual information bound can outperform comparable bounds in the literature by an arbitrarily large factor.

IEEE2019