Dirichlet Prior

The Dirichlet distribution is the conjugate prior to the multinomial distribution. This is a critical computational property: if the prior is Dirichlet and the likelihood is multinomial, the posterior distribution is also a Dirichlet distribution. This conjugacy enables efficient Gibbs sampling and variational inference in models like LDA, as the posterior parameters can be updated analytically by simply adding observed counts to the prior pseudo-counts.

The concentration parameter (α) dictates the shape of the Dirichlet distribution and, consequently, the sparsity of document-topic mixtures.

• α < 1: The distribution is pushed toward the corners of the simplex, enforcing sparsity. Documents are forced to be composed of very few topics.
• α = 1: Results in a uniform distribution over the simplex. All topic mixtures are equally likely.
• α > 1: The distribution concentrates toward the center, encouraging dense, uniform mixtures where documents contain many topics in similar proportions.

The Dirichlet distribution is defined over the probability simplex—a geometric space where every point is a valid probability vector (non-negative components summing to 1). For a model with K topics, the Dirichlet prior lives on a (K-1)-dimensional simplex. A single sample from a Dirichlet prior is a vector of length K that sums to exactly 1, representing a valid distribution over topics for a document.

The Dirichlet prior can be configured in two ways:

• Symmetric Prior: A single scalar α value is used for all dimensions, assuming no a priori preference for any topic. This is the standard setting in vanilla LDA.
• Asymmetric Prior: A vector of distinct α values is specified, encoding prior beliefs that some topics are more prevalent than others. This is used in Seeded LDA or when incorporating domain knowledge to guide the model toward specific thematic structures.

The Dirichlet distribution is the multivariate generalization of the Beta distribution. Just as the Beta distribution models a single probability value (e.g., the bias of a coin) on the interval [0,1], the Dirichlet models a vector of probabilities that sum to 1. For K=2, the Dirichlet distribution reduces exactly to a Beta distribution. This connection provides an intuitive bridge from simple binary uncertainty to complex categorical uncertainty.

In Bayesian inference, the Dirichlet prior acts as a natural regularizer that prevents overfitting. By adding pseudo-counts (the α values) to the observed data, the prior smooths the posterior estimates. This is particularly important in topic modeling where word co-occurrence data is sparse. Without this regularization, maximum likelihood estimates would assign zero probability to unseen word-topic combinations, breaking the model's ability to generalize to new documents.