Normalizing constant

The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.

Table of contents

1 Definition and examples 2 Bayes' theorem 3 Non-probabilitistic uses

Definition and examples

In probability theory, a normalizing constant is a constant by which an everywhere nonnegative function must be multiplied in order to get a probability density function or a probability mass function. For example, we have

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.)

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.

Bayes' theorem

In Bayes' theorem says that the product of the prior probability measure and the likelihood function is proportional to the posterior probability measure. Proportional to implies that one must multiply 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

where P(H₀) is the prior probability that the hypothesis is true; P(D|H₀) is the likelihood of the data given that the hypothesis is true; and 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:

.

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-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.