posteriorverwachte

Posteriorverwachte, in English the posterior expectation, is a central concept in Bayesian statistics. It is the expected value of a quantity with respect to the posterior distribution after observing data. Formally, if θ is a parameter with prior π(θ) and the data D have likelihood L(D|θ), the posterior density is p(θ|D) ∝ L(D|θ) π(θ). The posterior expected value of θ is E[θ|D] = ∫ θ p(θ|D) dθ (or ∑ θ p(θ|D) in the discrete case). More generally, for any function g(θ), the posterior expectation is E[g(θ)|D] = ∫ g(θ) p(θ|D) dθ.

The posterior expectation is widely used in estimation and decision theory. Under squared error loss, the Bayes estimator is the posterior mean E[θ|D]. Different loss functions yield other estimators, such as the posterior median under absolute error loss. The posterior expectation also serves to summarize uncertainty and to propagate it through downstream calculations, e.g., E[g(θ)|D] for a predictive or decision-related quantity.

Computation can be challenging when the posterior is not available in closed form. In practice, E[θ|D] is approximated via quadrature in simple models or via numerical methods such as Monte Carlo integration, using samples from p(θ|D) obtained by MCMC, importance sampling, or variational approximations.

Posteriorverwachte differs from the prior expectation E[θ], which conditions only on the prior and ignores data. It extends naturally to multivariate cases and to expectations of functions of several parameters.