Articles containing proofs | Theory of computation | Computability theory | Algorithmic information theory | Information theory

Chain rule for Kolmogorov complexity

The chain rule for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: That is, the combined randomness of two sequences X and Y is the sum of the randomness of X plus whatever randomness is left in Y once we know X.This follows immediately from the definitions of conditional and joint entropy, and the fact from probability theory that the joint probability is the product of the marginal and conditional probability: The equivalent statement for Kolmogorov complexity does not hold exactly; it is true only up to a logarithmic term: (An exact version, KP(x, y) = KP(x) + KP(y|x*) + O(1),holds for the prefix complexity KP, where x* is a shortest program for x.) It states that the shortest program printing X and Y is obtained by concatenating a shortest program printing X with a program printing Y given X, plus at most a logarithmic factor. The results implies that algorithmic mutual information, an analogue of mutual information for Kolmogorov complexity is symmetric: I(x:y) = I(y:x) + O(log K(x,y)) for all x,y. (Wikipedia).

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Even More Chain Rule

Even more examples using the chain rule.

From playlist Calculus

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This is How You Use the Chain Rule in Calculus

This is How You Use the Chain Rule in Calculus

From playlist Random calculus problems:)

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The Chain Rule: Part 1 of 2

http://mathispower4u.wordpress.com/

From playlist Differentiation Using the Chain Rule

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Chain rule for functions of two variables

Free ebook http://tinyurl.com/EngMathYT A example on the mathematics of the chain rule for functions of two variables.

From playlist A second course in university calculus.

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Kolmogorov Complexity - Applied Cryptography

This video is part of an online course, Applied Cryptography. Check out the course here: https://www.udacity.com/course/cs387.

From playlist Applied Cryptography

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Chain Rule for Several Variable Functions

How to apply the chain rule for partial deriviatves. An example is discussed. Free ebook tinyurl.com/EngMathYT

From playlist Several Variable Calculus / Vector Calculus

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Proof - The Chain Rule of Differentiation

This video proves the chain rule of differentiation. http://mathispower4u.com

From playlist Calculus Proofs

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Nexus Trimester - Andrei Romashchenko (LIRMM)

On Parallels Between Shannon’s and Kolmogorov’s Information Theories (where the parallelism fails and why) Andrei Romashchenko (LIRMM) February 02, 2016 Abstract: Two versions of information theory - the theory of Shannon's entropy and the theory of Kolmgorov complexity - have manifest

From playlist Nexus Trimester - 2016 - Distributed Computation and Communication Theme

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Gérard Letac: Quasi logistic distributions and Gaussian scale mixing

CIRM VIRTUAL EVENT Recorded during the meeting "Mathematical Methods of Modern Statistics 2" the June 02, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicia

From playlist Virtual Conference

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Stéphane Jaffard - Conférence organisée par l'Institut Fourier et le Laboratoire Jean Kuntzmann

Conférence organisée par l'Institut Fourier et le Laboratoire Jean Kuntzmann Licence: CC BY NC-ND 4.0

From playlist Conférences grand public "MathEnVille"

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Silke Glas: Symplectic model reduction of Hamiltonian systems on nonlinear manifolds

CONFERENCE Recorded during the meeting "Energy-Based Modeling, Simulation, and Control of Complex Constrained Multiphysical Systems" the April 19, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other

From playlist Numerical Analysis and Scientific Computing

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Pierre Baudot (8/19/20): Cohomological characterization of information structures

Speaker: Pierre Baudot, Median Technologies. In collaboration in part with Daniel Bennequin, Monica Tapia, and Jean-Marc Goaillard Title: Cohomological characterization of information and higher order statistical structures - Machine learning and statistical physics aspects Abstract: We

From playlist AATRN 2020

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The Chain Rule

Explanation of the Chain Rule In this video, I explain the chain rule, and illustrate it with a couple of examples. And at the end, I give an intuitive explanation of why the chain rule should be true. Who runs faster? Usain Bolt or a train? Watch this video to find out! Subscribe to my

From playlist Calculus

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Spotlight Talks - Amir Asadi, Dimitris Kalimeris

Workshop on Theory of Deep Learning: Where next? Topic: Spotlight Talks Speaker: Amir Asadi, Dimitris Kalimeris Date: October 15, 2019 For more video please visit http://video.ias.edu

From playlist Mathematics

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Andreï Kolmogorov: un grand mathématicien au coeur d'un siècle tourmenté

Conférence grand public au CIRM Luminy Andreï Kolmogorov est un mathématicien russe (1903-1987) qui a apporté des contributions frappantes en théorie des probabilités, théorie ergodique, turbulence, mécanique classique, logique mathématique, topologie, théorie algorithmique de l'informati

From playlist OUTREACH - GRAND PUBLIC

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Berry's Paradox - An Algorithm For Truth

Go to https://expressvpn.com/upandatom and find out how you can get 3 months free. Hi! I'm Jade. If you'd like to consider supporting Up and Atom, head over to my Patreon page :) https://www.patreon.com/upandatom Visit the Up and Atom store https://store.nebula.app/collections/up-and-at

From playlist Math

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Shannon 100 - 27/10/2016 - Jean Louis DESSALLES

Information, simplicité et pertinence Jean-Louis Dessalles (Télécom ParisTech) Claude Shannon fonda la notion d’information sur l’idée de surprise, mesurée comme l’inverse de la probabilité (en bits). Sa définition a permis la révolution des télécommunications numériques. En revanche, l’

From playlist Shannon 100

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Chain Rule Chain Rule Chain Rule

A statement of the chain rule, plus examples

From playlist Exam 2 Fall 2013, MAT 241

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Chain rule for functions of two variables

Free ebook http://tinyurl.com/EngMathYT A lecture on the mathematics of the chain rule for functions of two variables. Plenty of examples are presented to illustrate the ideas. These concepts are seen at university.

From playlist A second course in university calculus.

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Asymptotic efficiency in high-dimensional covariance estimation – V. Koltchinskii – ICM2018

Probability and Statistics Invited Lecture 12.18 Asymptotic efficiency in high-dimensional covariance estimation Vladimir Koltchinskii Abstract: We discuss recent results on asymptotically efficient estimation of smooth functionals of covariance operator Σ of a mean zero Gaussian random

From playlist Probability and Statistics

Related pages

Probability theory | Kolmogorov complexity | Joint entropy | Mutual information | Conditional probability | Randomness | Logarithm | Conditional entropy