Dirichletjakaumaa

Dirichlet distribution is a continuous multivariate probability distribution parameterized by a vector of positive real numbers. It is a generalization of the beta distribution to multiple dimensions. The Dirichlet distribution is commonly used in Bayesian statistics, machine learning, and other fields where it serves as a prior distribution for the parameters of a multinomial distribution.

The probability density function (PDF) of a Dirichlet distribution with parameters alpha_1, alpha_2, ..., alpha_K (where K is the number of dimensions) is given by:

f(x_1, ..., x_K; alpha_1, ..., alpha_K) = (Gamma(alpha_1 + ... + alpha_K) / (Gamma(alpha_1) * ... * Gamma(alpha_K))) * (x_1^(alpha_1 - 1) * ... * x_K^(alpha_K - 1))

where x_1, ..., x_K are non-negative real numbers that sum to 1, and Gamma is the gamma function.

The Dirichlet distribution has several useful properties. For example, the mean of a Dirichlet-distributed random variable is given by the vector of parameters divided by their sum. Additionally, the Dirichlet distribution is conjugate to the multinomial distribution, meaning that if the prior distribution of a parameter is Dirichlet, and the likelihood is multinomial, then the posterior distribution is also Dirichlet.

The Dirichlet distribution is named after the German mathematician Peter Gustav Lejeune Dirichlet, who first studied the distribution in the context of Fourier series.