Marginale Wahrscheinlichkeit is a statistical term that represents the probability of observing a set of data given a specific model, while accounting for all possible values of the model’s parameters. It is often denoted as P(Data | Model) and plays a crucial role in Bayesianischer Statistik.
In der Bayesschen analysis, we typically start with a prior distribution that reflects our beliefs about the parameters before observing any data. After observing the data, we update our beliefs to obtain a posterior distribution using Bayes’ theorem. However, to compute this posterior, we need to know the marginal likelihood.
The marginal likelihood is calculated by integrating the likelihood of the data given the parameters multiplied by the prior distribution of the parameters over the entire Parameterraum:
P(Daten | Modell) = ∫ P(Daten | Parameter) × P(Parameter | Modell) d(Parameter)
This integral sums the likelihood of the observed data across all possible parameter settings, weighing them by how plausible those parameters are according to the prior. Marginal likelihood is particularly useful for Modellvergleich, where we can compute the marginal likelihood for different models and use these values to determine which model is more likely given the data.
However, calculating marginal likelihood can be challenging, especially in high-dimensional parameter spaces, leading to the use of various approximation methods such as Monte-Carlo-Integration or Bayesian model averaging. Overall, marginal likelihood is a fundamental concept in Bayesian inference that helps statisticians and data scientists evaluate and compare models based on observed data.