Dirichlet distribution - Wikipedia

Abstraction: Multivariate continuous distribution over probability simplices used as Bayesian prior

Key points:

• Parameterized by vector alpha of positive reals; support is the open (K-1)-simplex — i.e., distributions over K categories
• Conjugate prior of the categorical and multinomial distributions: if prior is Dir(alpha) and we observe counts c, posterior is Dir(alpha + c), enabling sequential updating
• Concentration parameter alpha controls sparsity: alpha < 1 produces sparse samples (mass concentrated in few categories); alpha > 1 produces dense, nearly uniform samples; alpha = 1 is flat/uniform Dirichlet
• Sampling algorithm: draw K independent Gamma(alpha_i, 1) variates, then normalize by their sum
• Widely used in Bayesian mixture models, hierarchical models, and topic modeling (e.g., LDA) as prior over topic/category proportions
• Infinite-dimensional generalization is the Dirichlet process

Connections: Dirichlet Distribution · Bayesian Inference · Conjugate Prior · Mixture Models

Source: http://en.wikipedia.org/wiki/Dirichlet_distribution