# Posterior probability

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The posterior probability of a random event or an uncertain proposition is the conditional probability it is assigned when the relevant evidence is taken into account.

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

$f_{X\mid Y=y}(x)={f_X(x) L_{X\mid Y=y}(x) \over {\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx}}$

gives the posterior probability density function for a random variable X given the data Y = y, where

• $f_X(x)$ is the prior density of X,
• $L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y)$ is the likelihood function as a function of x,
• $\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx$ is the normalizing constant, and
• $f_{X\mid Y=y}(x)$ is the posterior density of X given the data Y = y.
Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound PoissondegeneratedegreeGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normalvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropy phase-typeposterior priorquasisampling
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