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{{Expert-subject|Mathematics|date=November 2008}} | |||
In [[information theory]], the '''limiting density of discrete points''' is an adjustment to the formula of [[Claude Shannon]] for [[differential entropy]]. | |||
It was formulated by [[Edwin Thompson Jaynes]] to address defects in the initial definition of differential entropy. | |||
== Definition == | |||
Shannon originally wrote down the following formula for the [[entropy]] of a continuous distribution, known as '''differential entropy''': | |||
: <math>H(X)=-\int p(x)\log p(x)\,dx.</math> | |||
Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral) and it turns out to lack many of the properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a [[change of variables]] and can even become negative. | |||
Jaynes (1963, 1968) argued that the formula for the continuous entropy should be derived by taking the limit of increasingly dense discrete distributions<ref>{{cite book|authorlink=Edwin Thompson Jaynes|author=Jaynes, E. T.|url=http://bayes.wustl.edu/etj/articles/brandeis.pdf|year=1963|chapter=Information Theory and Statistical Mechanics|title=Statistical Physics|editor=K. Ford|publisher=Benjamin, New York|pages=181}}</ref><ref>{{cite journal|author=Jaynes, E. T.|year=1968|title=Prior Probabilities|journal=IEEE Trans. on Systems Science and Cybernetics|volume=SSC-4|pages=227|url=http://bayes.wustl.edu/etj/articles/prior.pdf}}</ref>. Suppose that we have a set of <math>n</math> discrete points <math>\{x_i\}</math>, such that in the limit <math>n \to \infty</math> their density approaches a function <math>m(x)</math> called the [[invariant measure]]. | |||
:<math> \lim_{n \to \infty}\frac{1}{n}\,(\mbox{number of points in }a<x<b)=\int_a^b m(x)\,dx</math> | |||
Jaynes derived from this the following formula for the continuous entropy, which he argued should be taken as the correct formula: | |||
: <math>H(X)=-\int p(x)\log\frac{p(x)}{m(x)}\,dx.</math> | |||
It is formally similar to but conceptually distinct from the (negative of the) [[Kullback–Leibler divergence]] or relative entropy, which is comparison measure between two probability distributions. The formula for the Kullback-Leibler divergence is similar, except that the invariant measure <math>m(x)</math> is replaced by a second probability density function <math>q(x)</math>. In Jaynes' formula, <math>m(x)</math> is not a probability density but simply a density. In particular it does not have to be normalised so as to sum to 1. | |||
Jaynes' continuous entropy formula and the relative entropy share the property of being invariant under a change of variables, which solves many of the difficulties that come from applying Shannon's continuous entropy formula. | |||
It is the inclusion of the invariant measure <math>m</math> that ensures the formula's invariance under a change of variables, since both <math>m(x)</math> and <math>p(x)</math> must be transformed in the same way. | |||
==References== | |||
{{reflist}} | |||
* {{cite book|author=Jaynes, E. T.|year=2003|title=Probability Theory: The Logic of Science|publisher=Cambridge University Press|isbn=978-0521592710}} | |||
{{DEFAULTSORT:Limiting Density Of Discrete Points}} | |||
[[Category:Probability theory]] | |||
[[Category:Theory of probability distributions]] | |||
[[Category:Information theory]] | |||
{{probability-stub}} | |||
Revision as of 01:30, 30 November 2013
Template:Expert-subject In information theory, the limiting density of discrete points is an adjustment to the formula of Claude Shannon for differential entropy.
It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differential entropy.
Definition
Shannon originally wrote down the following formula for the entropy of a continuous distribution, known as differential entropy:
Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral) and it turns out to lack many of the properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a change of variables and can even become negative.
Jaynes (1963, 1968) argued that the formula for the continuous entropy should be derived by taking the limit of increasingly dense discrete distributions[1][2]. Suppose that we have a set of discrete points , such that in the limit their density approaches a function called the invariant measure.
Jaynes derived from this the following formula for the continuous entropy, which he argued should be taken as the correct formula:
It is formally similar to but conceptually distinct from the (negative of the) Kullback–Leibler divergence or relative entropy, which is comparison measure between two probability distributions. The formula for the Kullback-Leibler divergence is similar, except that the invariant measure is replaced by a second probability density function . In Jaynes' formula, is not a probability density but simply a density. In particular it does not have to be normalised so as to sum to 1.
Jaynes' continuous entropy formula and the relative entropy share the property of being invariant under a change of variables, which solves many of the difficulties that come from applying Shannon's continuous entropy formula.
It is the inclusion of the invariant measure that ensures the formula's invariance under a change of variables, since both and must be transformed in the same way.
References
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