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2005-12-29T22:54:54Z

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The '''posterior probability''' can be calculated by [[Bayes' theorem]] from the prior probability and the [[likelihood function]].

Similarly a '''posterior probability distribution''' is the conditional [[probability distribution]] of the uncertain quantity given the data. It can be calculated by multiplying the [[prior probability distribution]] by the [[likelihood function]], and then dividing by the [[normalizing constant]]. For example

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

gives the posterior [[probability density function]] for a random variable ''X'' given the data ''Y''=''y'', where
* <math>f_X(x)</math> is the prior density of X,
* <math>L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y)</math> is the likelihood function as a function of ''x'',
* <math>\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx</math> is the normalizing constant, and
* <math>f_{X\mid Y=y}(x)</math> is the posterior density of ''X''.

==External links==
* [http://en.wikipedia.org/wiki/Posterior_probability Wikipedia article on '''Posterior probability''']

[[Category:Academic Research]]
[[Category:Statistics]]

Posterior probability - Revision history

Christoph: Started article