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Definition and examples[édit | sunting sumber]
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
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[édit | sunting sumber]
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 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[édit | sunting sumber]
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.