Dirichlet distribution

A probability distribution over vectors of nonnegative proportions that sum to one. Its concentration parameters control how probability mass is distributed across the simplex.

Simplex: A geometric space formed by nonnegative coordinates whose sum is one. The Dirichlet distribution assigns probability across this space of possible proportion vectors.
Probability simplex: The set of probability vectors whose entries are nonnegative and sum to one. It is the sample space on which Dirichlet-distributed vectors live.
Bayesian inference: A framework for updating probability distributions over unknown quantities using observed data. Dirichlet priors make Bayesian updates for category probabilities especially simple.
Logistic-normal distribution: A distribution on proportions formed by transforming a multivariate normal vector with the softmax function. Unlike the Dirichlet, it can represent richer dependence among log-ratios of proportions.
Concentration parameter: A parameter controlling how concentrated a probability distribution is around its central values. The sum of the Dirichlet parameters governs dispersion around the mean proportions.
Probability density function: A function whose integral over a region gives the probability of a continuous random variable falling there. The Dirichlet density specifies relative probability across the simplex.
Multinomial distribution: A distribution for counts of outcomes across categories in repeated independent trials with fixed probabilities. A Dirichlet prior on its category probabilities produces a tractable posterior.
Dirichlet process: A distribution over probability measures whose finite measurable partitions have Dirichlet-distributed probabilities. It generalizes Dirichlet modeling from a fixed finite vector to random measures.
Gamma distribution: A continuous probability distribution on positive real numbers, parameterized by shape and scale. Independent gamma draws, normalized by their sum, generate Dirichlet-distributed proportions.
Conjugate prior: A prior distribution that yields a posterior in the same distribution family after observing data. The Dirichlet is conjugate to the categorical and multinomial likelihoods.

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