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.

Sampling and Estimation

Generating random samples from a Dirichlet distribution is typically accomplished by drawing Gamma-distributed variables for each component and normalizing them so they sum to one. This approach underpins many Bayesian models where proportions evolve through stochastic simulation. When fitting a Dirichlet to data, a common method is maximum likelihood estimation using iterative algorithms such as fixed-point iteration or Newton’s method. These techniques adjust the $\alpha$ parameters so the predicted mean proportions align with observed frequencies.

Connections to Other Distributions

The Dirichlet is closely related to the beta distribution, which describes probabilities for a single proportion. In fact, if you fix all but two components at zero, the Dirichlet reduces to a beta. Its role as a conjugate prior for the multinomial distribution means posterior updates are straightforward: new observations simply add to the existing shape parameters. This property makes the Dirichlet a foundational tool in Bayesian text modeling, genetic studies, and anywhere else where categorical counts are analyzed.

Moments and Interpretations

The expected value of each component is the corresponding shape parameter divided by the sum of all parameters. In symbols, the mean vector is $(\frac{\alpha }{1_{}},\frac{\alpha }{2_{}},\frac{\alpha }{3_{}})$. These values sit at the center of mass of the simplex, clarifying how the parameters serve as pseudo-counts: increasing $\alpha 1_{}$ pulls probability toward the first vertex just as recording additional observations of that outcome would. When each $\alpha$ exceeds one, the distribution possesses an interior mode at $(\frac{\alpha }{1_{}}\Sigma \alpha -3,\frac{\alpha }{2_{}}\Sigma \alpha -3,\frac{\alpha }{3_{}}\Sigma \alpha -3)$, highlighting the most probable composition.

Geometric Perspective

Visualizing the Dirichlet distribution involves a two-dimensional simplex—an equilateral triangle—when there are three categories. Each vertex corresponds to allocating all probability to one category. Lines parallel to a side represent constant values of the opposite component. By shading regions according to density, one can see how different parameter choices warp the distribution across this triangle. Equal parameters produce symmetric, centrally concentrated contours, whereas a dominant parameter stretches the high-density region toward its vertex. This geometric picture aids intuition for those new to multivariate probabilities.

Real-World Examples

Consider a survey in which respondents divide their time among work, leisure, and sleep. If previous studies suggest an average split of 40%, 30%, and 30%, you might encode this belief with $\alpha$ parameters proportional to those percentages, perhaps scaled to reflect confidence—$(4,3,3)$ for moderate certainty or $(40,30,30)$ for strong prior data. As new survey results arrive, simply add the observed counts to the respective parameters to obtain a revised distribution representing updated beliefs about how people allocate their day. In machine learning, topic models such as Latent Dirichlet Allocation treat the $\alpha$ vector as a prior over document-topic mixtures; tweaking these values influences whether documents concentrate on a few topics or spread their content widely.

Working Through an Example

Suppose $(\alpha \_1,\alpha \_2,\alpha \_3)=(2,5,3)$ to represent a prior belief about proportions of three species in an ecosystem. To evaluate how plausible the composition $(x\_1,x\_2,x\_3)=(0.1,0.6,0.3)$ is under this model, enter the parameters and coordinates into the calculator. The PDF output might be around 3.5, indicating a relatively typical mixture given the prior counts. The mean of this distribution is $(0.2,0.5,0.3)$, slightly different from the evaluated point but still close. Because each α is greater than one, a mode exists at approximately $(0.125,0.5625,0.3125)$. Analysts can compare the observed point to both the mean and mode to gauge how strongly it agrees with prior expectations.

Practical Tips and Caveats

While the Dirichlet distribution offers mathematical convenience, it also carries assumptions. In particular, it enforces a form of negative correlation: increasing one component necessarily decreases the others. For some applications this mirrors reality, but for others, a more flexible model such as the logistic-normal distribution might be appropriate. Additionally, large α values concentrate the distribution tightly around the mean, which may or may not reflect true variability in the system being modeled. Always examine sensitivity to parameter choices, especially when the prior is subjective.

Further Reading

If you wish to dig deeper, textbooks on Bayesian data analysis cover the Dirichlet distribution extensively, often alongside multinomial models. Visual demonstrations of the simplex can be found in many online resources, and interactive plots reveal how changing α shifts the density landscape. Gaining intuition for these shapes enhances your ability to construct informative priors and interpret compositional data responsibly.