Gibbs Sampling ist eine Markov-Ketten-Monte-Carlo (MCMC) algorithm used for obtaining a sequence of samples from a Gemeinsame Wahrscheinlichkeitsverteilung when direct sampling is difficult. This technique is particularly useful in high-dimensional spaces where traditional sampling methods may fail.
The core idea behind Gibbs Sampling is to iteratively sample from the conditional distributions of each variable, given the current values of all the other variables. For instance, in a scenario with two variables, X and Y, the algorithm would first sample a value for X from its conditional distribution P(X|Y), then update Y by sampling from P(Y|X), and repeat this process. This results in a sequence of samples that converge to the target gemeinsame Verteilung.
One of the key advantages of Gibbs Sampling is its simplicity and ease of implementation, especially when the conditional distributions are easy to sample from. It is widely used in various fields such as Bayesianischer Statistik, machine learning, and image processing. However, Gibbs Sampling can have slow convergence rates, particularly if the variables are highly correlated.
In der Praxis werden die ersten paar Stichproben, die durch Gibbs Sampling erzeugt werden, oft verworfen (ein Prozess, der als Burn-in bekannt ist), um der Kette zu ermöglichen, gegen die Zielverteilung zu konvergieren. Die verbleibenden Stichproben können dann verwendet werden, um Eigenschaften der Verteilung zu schätzen, wie Mittelwerte, Varianzen und andere Statistiken.