What is the Dirichlet distribution?

The Dirichlet distribution is a probability distribution over a set of probabilities.

The Dirichlet distribution is a multivariate probability distribution that is commonly used in Bayesian statistics to model the distribution of probabilities over a set of events. It is a generalization of the beta distribution, which is a probability distribution over a single probability. The Dirichlet distribution is defined over a set of K probabilities, where K is a positive integer. The distribution is parameterized by a vector α of K positive real numbers, which is called the concentration parameter. The concentration parameter determines the shape of the distribution, with larger values of α resulting in more concentrated distributions.

The probability density function of the Dirichlet distribution is given by:

f(x_1, ..., x_K | α_1, ..., α_K) = (1 / B(α)) * ∏_{i=1}^K x_i^(α_i - 1)

where x_1, ..., x_K are the probabilities, α_1, ..., α_K are the concentration parameters, and B(α) is the multivariate beta function, which is defined as:

B(α) = ∏_{i=1}^K Γ(α_i) / Γ(∑_{i=1}^K α_i)

where Γ is the gamma function.

The Dirichlet distribution has several important properties, including the fact that the sum of the probabilities is always equal to 1, and that the mean and variance of the distribution can be expressed in terms of the concentration parameter. The Dirichlet distribution is widely used in Bayesian statistics for modelling the distribution of probabilities over a set of events, such as the probabilities of different outcomes in a game or the probabilities of different topics in a corpus of documents.