posteriorprädiktive

Posteriorprädiktive, in Bayesian statistics often referred to as the posterior predictive distribution, describes the distribution of future observations given observed data under a specified model. It is obtained by averaging the likelihood of new data over the uncertainty about the model parameters as quantified by their posterior distribution.

Formally, for a future observation y* and data y under a model with parameter θ, the posteriorprädiktive Verteilung is p(y* | y) = ∫ p(y* | θ) p(θ | y) dθ. If multiple future observations are considered, the joint posterior predictive distribution is p(y*1, ..., y*m | y) = ∫ ∏ p(y*i | θ) p(θ | y) dθ. The posterior distribution p(θ | y) itself is obtained through Bayes' theorem: p(θ | y) ∝ p(y | θ) p(θ).

This distribution reflects both aleatoric uncertainty (inherent randomness of data given θ) and epistemic uncertainty (uncertainty about θ given the observed data). It is central to predictive inference, forecasting, and model checking. Posterior predictive checks compare observed data to samples drawn from p(y* | y) to assess whether the model can reproduce salient features of the data. The approach contrasts with the prior predictive distribution, which integrates over the prior p(θ) rather than the posterior, thereby incorporating only prior uncertainty.

Computation typically relies on sampling methods. When the posterior p(θ | y) is available in closed form, analytical integration may be possible. Otherwise, one draws θ^(s) ~ p(θ | y) and, for each draw, samples y*^(s) ~ p(y* | θ^(s)) to approximate the posterior predictive distribution, often via Markov chain Monte Carlo or variational techniques.

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