PosteriorFunktion

PosteriorFunktion is a term used in Bayesian statistics to refer to the posterior distribution or the posterior density function of a parameter after observing data. It is derived from Bayes' theorem and combines prior information with the likelihood of the observed data. If D denotes the data and θ a parameter, the posterior density is p(θ|D) = p(D|θ) p(θ) / p(D), where p(D) is the marginal likelihood obtained by integrating the joint density over all θ. The posterior is defined up to this normalization constant and represents updated beliefs about θ after seeing D.

The posterior encodes both the prior beliefs and the information gained from the data. It allows the computation of summary statistics such as the posterior mean, posterior median, or posterior mode (also called the MAP estimate) and provides a measure of uncertainty through the posterior variance or credible intervals. In some models, the posterior has a closed form, a property known as conjugacy; common examples include the normal–normal, beta–binomial, and gamma–Poisson pairs. In many practical models, the posterior does not have a closed form and numerical methods are used.

Computational methods for the PosteriorFunktion include Markov chain Monte Carlo (MCMC) to sample from the posterior, importance sampling, and variational inference to approximate it. The posterior predictive distribution, obtained by integrating the likelihood over the posterior, is often used for predicting new data. The PosteriorFunktion is a central concept in Bayesian inference and underpins parameter estimation, uncertainty quantification, and model comparison.