The Dirichlet Distribution

The Dirichlet distribution generates random vectors whose entries are all positive and sum exactly to one — a natural fit whenever you're modeling a set of proportions, like portfolio weights across assets or the probability of each outcome in a multi-way classification. It's the multi-dimensional generalization of the Beta distribution: where a Beta distribution describes uncertainty about a single probability (like a coin's bias), a Dirichlet describes uncertainty about several probabilities at once that must add up to one, such as the chance a trade executes at each of several possible venues.

It's controlled by a vector of concentration parameters $(\alpha_1, \ldots, \alpha_k)$, one per category. Equal, large values (e.g., all $\alpha_i = 50$) produce draws clustered tightly around equal proportions; equal, small values (e.g., all $\alpha_i = 0.5$) produce draws that tend toward extreme, corner-heavy splits, with most of the probability mass on just one or two categories; and unequal values shift the average draw toward the categories with larger $\alpha_i$. For example, with $\alpha = (10, 10, 10)$ for three asset classes, a typical draw might be weights like (0.36, 0.31, 0.33) — close to equal, with only modest spread — whereas $\alpha = (1, 1, 1)$ can easily produce something like (0.05, 0.75, 0.20).

Its most common quant use is as a Bayesian prior over category probabilities: given observed counts of outcomes (which venue filled an order, which regime the market was in), a Dirichlet prior updates into another Dirichlet posterior after seeing data, the same conjugate-update convenience that makes the Beta distribution popular for a single probability.