Gibbs Samplingは マルコフ連鎖モンテカルロ (MCMC) algorithm used for obtaining a sequence of samples from a 結合確率分布から 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 結合分布.
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 ベイズ統計学, machine learning, and image processing. However, Gibbs Sampling can have slow convergence rates, particularly if the variables are highly correlated.
実際には、ギブスサンプリングによって生成される最初の数サンプルは、チェーンがターゲット分布に収束するのを待つために破棄されることが多いです（バーンインと呼ばれるプロセス）。残りのサンプルは、平均や分散などの分布の性質を推定するために使用されます。