Editing Posterior probability (section)

== Calculation ==

The posterior probability distribution of one [[random variable]] given the value of another can be calculated with [[Bayes' theorem]] by multiplying the [[prior probability distribution]] by the [[likelihood function]], and then dividing by the [[normalizing constant]], as follows:

:<math>f_{X\mid Y=y}(x)={f_X(x) \mathcal L_{X\mid Y=y}(x) \over {\int_{-\infty}^\infty f_X(u) \mathcal L_{X\mid Y=y}(u)\,du}}</math>

gives the posterior [[probability density function]] for a random variable <math>X</math> given the data <math>Y=y</math>, where

* <math>f_X(x)</math> is the prior density of <math>X</math>,
* <math>\mathcal L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y)</math> is the likelihood function as a function of <math>x</math>,
* <math>\int_{-\infty}^\infty f_X(u) \mathcal L_{X\mid Y=y}(u)\,du</math> is the normalizing constant, and
* <math>f_{X\mid Y=y}(x)</math> is the posterior density of <math>X</math> given the data <math>Y=y</math>.<ref>{{Cite web |title=Posterior probability - formulasearchengine |url=https://formulasearchengine.com/wiki/Posterior_probability |access-date=2022-08-19 |website=formulasearchengine.com}}</ref>