Normalizing constant

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Konsep ngeunaan normalizing constant ningkat dina probability theory jeung dina widang matematik sejenna.

Definition and examples

Dina tiori probabiliti, normalisasi konstanta nyaeta konstanta nu di unggal tempat fungsi non negatip kudu dikalikeun dina usaha keur meunangkeun fungsi probabiliti densiti atawa fungsi probabiliti masa. Contona, urang mibanda

$\int _{-\infty }^{\infty }e^{-x^{2}/2}\,dx={\sqrt {2\pi \,}},$

mangka

$\varphi (x)={\frac {1}{\sqrt {2\pi \,}}}e^{-x^{2}/2}$

ngarupakeun fungsi densiti probabiliti. Hal ieu ngarupakeun densiti standar sebaran normal. (Standar, dina kasus ieu hartina nilai ekspektasi sarua jeung 0 sarta varian sarua jeung 1.)

Similarly,

$\sum _{n=0}^{\infty }{\frac {\lambda ^{n}}{n!}}=e^{\lambda },$

and consequently

$f(n)={\frac {\lambda ^{n}e^{-\lambda }}{n!}}$

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.

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.

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

$P(H_{0}|D)={\frac {P(D|H_{0})P(H_{0})}{P(D)}}$

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:

$P(H_{0}|D)\sim P(D|H_{0})P(H_{0})$.

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

$P(H_{0}|D)={\frac {P(D|H_{0})P(H_{0})}{\sum _{i}P(D|H_{i})P(H_{i})}}.$

In this case, the value

$P(D)=\sum _{i}P(D|H_{i})P(H_{i})\;$

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

Non-probabilitistic 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.