Dirichletdistributed

The Dirichlet distribution is a family of continuous multivariate probability distributions parameterized by a vector of positive real numbers. It is a generalization of the beta distribution to multiple dimensions. The Dirichlet distribution is often used as a prior distribution in Bayesian statistics, particularly for categorical or multinomial distributions.

The probability density function of a Dirichlet distribution with parameter vector $\alpha = (\alpha_1, \dots, \alpha_K)$ where $\alpha_i > 0$ for all $i$, is given by:

$f(x_1, \dots, x_K; \alpha_1, \dots, \alpha_K) = \frac{1}{B(\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}$

subject to the constraints $x_i \ge 0$ for all $i$ and $\sum_{i=1}^K x_i = 1$. Here, $B(\alpha)$ is the multivariate beta function, which acts as a normalization constant.

The Dirichlet distribution is conjugate to the multinomial distribution. This means that if the prior distribution of the parameters of a multinomial distribution is a Dirichlet distribution, then the posterior distribution of these parameters will also be a Dirichlet distribution. This conjugacy property makes the Dirichlet distribution very convenient for Bayesian inference.

The expected value of the $i$-th component of a Dirichlet distributed random vector is $E[X_i] = \frac{\alpha_i}{\sum_{j=1}^K \alpha_j}$. The variance and covariance structure of the Dirichlet distribution are also dependent on the parameters $\alpha$. When all $\alpha_i$ are equal, the distribution is symmetric. As the $\alpha_i$ values become large, the distribution becomes more concentrated around its mean.