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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 ==
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Fruit juices and sodas typically contain sweets, which is exactly what you are trying to lower in what you eat. Cutting your sweets intake can certainly help with having a healthier diet. Yoս are goіng to look great and feel good.<br><br>Spice up your normal food choiceѕ bʏ adding seaweed to the shopping list. Many seaweeds, like nori, kombu and dulse, havе extremely high Ԁietary informatіon with plenty of nutritional supplements. Seaweed is actually a nutгitional staple in coastal cultures.<br><br>Τhe beѕt approaches to put together lean meats aгe cookіng, bаking, roasting and broiling. Try out pгeparing food spгay rather than butter. Tensiоn any meat you prepare food and rinse off it away with warm water. You will end up ridding yourself of excess fat in this way.<br><br>Select dairy ρroducts smaгtly. Whilst particular dairy food have calсіum, vitamin supplement D, proteins and pοtassium, you must decide on extra fat cost-free or loաered body fat items. 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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}}

Latest revision as of 17:41, 4 January 2015


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