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Encyclopedia > Normalizing constant

The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. Probability theory is the mathematical study of probability. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...


Definition and examples

In probability theory, a normalizing constant is a constant by which an everywhere nonnegative function must be multiplied in order that the area under its graph is 1, i.e. it is a probability density function or a probability mass function. For example, we have Probability theory is the mathematical study of probability. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

so that

is a probability density function. This is the density of the standard normal distribution. (Standard, in this case, means the expected value is 0 and the variance is 1.) The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...


Similarly,

and consequently

is a probability mass function on the set of all nonnegative integers. This is the probability mass function of the Poisson distribution with expected value λ. In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...


The normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. In that context, the normalizing constant is called the partition function. The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...


Bayes' theorem


Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function . Proportional to implies that one must multiply or divide by a normalizing constant in order to assign measure 1 to the whole space, i.e., to get a probability measure. In a simple discrete case we have Bayes theorem is a result in probability theory. ... In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ...

where P(H0) is the prior probability that the hypothesis is true; P(D|H0) is the conditional probability of the data given that the hypothesis is true, but given that the data are known it is the likelihood of the hypothesis (or its parameters) given the data; P(H0|D) is the posterior probability that the hypothesis is true given the data. P(D) should be the probability of producing the data, but on its own is difficult to calculate, so an alternative way to describe this relationship is as one of proportionality: This article defines some terms which characterize probability distributions of two or more variables. ... In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ...

.

Since P(H|D) is a probability, the sum over all possible (mutually exclusive) hypotheses should be 1, leading to the conclusion that

In this case, the value

is the normalizing constant. It can be extended from countably many hypotheses to uncountably many by replacing the sum by an integral.


Non-probabilistic uses

The Legendre polynomials are characterized by orthogonality with respect to the uniform measure on the interval [− 1, 1] and the fact that they are normalized so that their value at 1 is 1. The constant by which one multiplies a polynomial in order that its value at 1 will be 1 is a normalizing constant. In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...


  Results from FactBites:
 
Normalizing constant - Wikipedia, the free encyclopedia (394 words)
In probability theory, a normalizing constant is a constant by which an everywhere nonnegative function must be multiplied in order that the area under its graph is 1, i.e.
In that context, the normalizing constant is called the partition function.
The constant by which one multiplies a polynomial in order that its value at 1 will be 1 is a normalizing constant.
  More results at FactBites »

 
 

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