Editing Normalizing constant (section)

==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 to assign measure 1 to the whole space, i.e., to get a probability measure.  In a simple discrete case we have
<math display="block">P(H_0|D) = \frac{P(D|H_0)P(H_0)}{P(D)}</math>
where P(H<sub>0</sub>) is the prior probability that the hypothesis is true; P(D|H<sub>0</sub>) is the [[conditional probability]] of the data given that the hypothesis is true, but given that the data are known it is the [[likelihood function|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 display="block">P(H_0|D) \propto P(D|H_0)P(H_0).</math>
Since P(H|D) is a probability, the sum over all possible (mutually exclusive) hypotheses should be 1, leading to the conclusion that
<math display="block">P(H_0|D) = \frac{P(D|H_0)P(H_0)}{\displaystyle\sum_i P(D|H_i)P(H_i)} .</math>
In this case, the [[Multiplicative inverse|reciprocal]] of the value
<math display="block">P(D) = \sum_i P(D|H_i)P(H_i) \;</math>
is the ''normalizing constant''.<ref>{{harvnb|Feller|1968|p=124}}</ref>  It can be extended from countably many hypotheses to uncountably many by replacing the sum by an integral.

For concreteness, there are many methods of estimating the normalizing constant for practical purposes. Methods include the bridge sampling technique, the naive Monte Carlo estimator, the generalized harmonic mean estimator, and importance sampling.<ref>{{Cite web |last = Gronau | first = Quentin | date = 2020 | title = bridgesampling: An R Package for Estimating Normalizing Constants | url = https://cran.r-project.org/web/packages/bridgesampling/vignettes/bridgesampling_paper.pdf | access-date = September 11, 2021 | website = The Comprehensive R Archive Network}}</ref>