Normalizing constant: Difference between revisions

Revision as of 12:03, 3 March 2009

Revision as of 18:43, 1 May 2009

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==Definition and examples==

In [[probability theory]], a '''normalizing constant''' is a constant by which an everywhere nonnegativenon-negative function must be multiplied in order thatso the area under its graph is 1, i.e.g., to make it a [[probability density function]] or a [[probability mass function]].<ref>''Continuous Distributions'' at University of Alabama.</ref><ref>Feller, 1968, p. 22.</ref> For example, we have

:<math>\int_{-\infty}^\infty e^{-x^2/2}\,dx=\sqrt{2\pi\,},</math>

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==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

:<math>P(H_0|D) = \frac{P(D|H_0)P(H_0)}{P(D)}</math>

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==Non-probabilistic uses==

The [[Legendre polynomials]] are characterized by [[orthogonality]] with respect to the uniform measure on the interval [&minus; 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 thatso its value at 1 will beis 1 is a normalizing constant.

[[Orthonormal]] functions are normalized such that