ディリクレ分布

ディリクレ分布

ディリクレ分布は連続的な 確率分布 defined over a simplex, which is a generalization of a triangle or tetrahedron in 高次元. It is primarily used in statistics and 機械学習 to model the distribution of proportions among multiple categories. Each category’s probability is represented as a component of a vector, and the Dirichlet distribution ensures that the sum of these probabilities equals one.

Formally, the Dirichlet distribution is parameterized by a vector of positive real numbers, often denoted as α = (α₁, α₂, ..., αₖ), where k is the number of categories. Each αᵢ serves as a concentration parameter that influences the distribution’s shape. A larger αᵢ indicates that the corresponding category is more likely to have a higher proportion, while smaller values suggest a lower likelihood.

The probability density function (PDF) of the Dirichlet distribution for a vector x = (x₁, x₂, ..., xₖ) is given by:

f(x; α) = (1/B(α)) * Π (xᵢ^(αᵢ - 1))

where B(α) is the multivariate Beta function, which normalizes the distribution, ensuring that the total probability is equal to one.

ディリクレ分布は広く使用されています ベイズ統計学, particularly as a prior distribution for multinomial distributions. It is also essential in various applications, including 自然言語処理, genetics, and any area where proportions of different categories need to be modeled.