The Multivariate Beta Family

The Dirichlet distribution generalizes the beta distribution to multiple proportions that sum to one. For parameters $\alpha 1_{}$, $\alpha 2_{}$, and $\alpha 3_{}$, the density over points $(x{1}_{},x{2}_{},x{3}_{})$ in the simplex $x{1}_{}+x{2}_{}+x{3}_{}=1$ is $f\left(x\right)=\frac{\Gamma \left(\alpha ,1_{},+,\alpha ,2_{},+,\alpha ,3_{}\right)}{\Gamma \left(\alpha ,1_{}\right)\Gamma \left(\alpha ,2_{}\right)\Gamma \left(\alpha ,3_{}\right)}{x}^{1_{}}{x}^{2_{}}{x}^{3_{}}$.

Each exponent is one less than its corresponding shape parameter, coupling the concentration of probability around the simplex corners. When all $\alpha$ values exceed one, the distribution favors interior regions. If some drop below one, it allocates more mass near the respective edges. The normalizing constant involves the multivariate beta function $B\left(\alpha 1_{},\alpha 2_{},\alpha 3_{}\right)$, ensuring the total probability integrates to one.

Applications span Bayesian statistics and compositional data analysis. For example, proportions of time spent in different activities, or the fraction of elements in a chemical mixture, naturally live in the simplex. The Dirichlet distribution provides a flexible prior over such proportions. Its conjugacy with the multinomial distribution allows convenient updating of beliefs when new counts are observed.

Using this calculator, you input three positive shape parameters and a candidate point whose coordinates sum to one. The script computes the PDF using gamma functions and simple arithmetic. In practice, you can interpret the result as the relative likelihood of seeing those proportions in a sample drawn from the modeled process.

Experiment by varying $\alpha$. When all values match, the distribution is symmetric. Increasing one parameter relative to the others biases the density toward that vertex. The Dirichlet thus captures prior knowledge about likely compositions.