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Definition and examplesEdit
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., to make it a probability density function or a probability mass function. For example, we have
Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. The parametrised 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 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
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:
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 reciprocal of the value
is the normalizing constant. It can be extended from countably many hypotheses to uncountably many by replacing the sum by an integral.
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.
Orthonormal functions are normalized such that
with respect to some inner product <f, g>.