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Normalizing Constant
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DEFINITION AND EXAMPLES


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. it is a Probability Density Function or a Probability Mass Function . For example, we have

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

so that

: arphi(x) = rac{1}{\sqrt{2\pi\,}} e^{-x^2/2}

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,

:\sum_{n=0}^\infty rac{\lambda^n}{n!}=e^\lambda ,

and consequently

:f(n)= rac{\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.

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


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(H<sub>0</sub>) Is The Prior Probability That The Hypothesis Is True P(DH<sub>0</sub>) Is The "http://wwwinformationdelightinfo/encyclopedia/entry/conditional_probability" class="copylinks">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(H<sub>0</sub>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:
  :<math>P(H 0D) rac{P(DH_0)P(H_0)}{\sum_i P(DH_i)P(H_i)} </math>
  :<math>P(D) \sum_i P(DH_i)P(H_i) \</math>