Dirichlet Distribution
X ~ Dirichlet(α)
Simplex / Conjugate-Prior Family • Published August 3, 2026
The conjugate prior for a probability vector, and the direct multivariate generalization of the already-published Beta distribution: every marginal and every aggregated grouping of categories is itself Beta or Dirichlet, an exact posterior-update tie to the already-published Multinomial sheet, a maximum-likelihood system with no closed form requiring digamma-based fixed-point iteration, and simulation via a ratio of independent Gamma draws.
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About This Reference Sheet
The conjugate prior for a probability vector, and the direct multivariate generalization of the already-published Beta distribution: every marginal and every aggregated grouping of categories is itself Beta or Dirichlet, an exact posterior-update tie to the already-published Multinomial sheet, a maximum-likelihood system with no closed form requiring digamma-based fixed-point iteration, and simulation via a ratio of independent Gamma draws.
Support
x = (x1,...,xk), xi > 0, Σxi = 1 (the simplex)
Parameters
α = (α1,...,αk), each αi > 0 (concentration parameters)