What the Dirichlet distribution calculator measures
The Dirichlet distribution models a three-part composition: positive shares that must add to 1. It is useful for market-share splits, vote shares among three options, topic proportions, portfolio weights, mixture fractions, and other settings where increasing one share necessarily leaves less for the other two. Every valid composition lies in a triangular region called the simplex.
This Dirichlet calculator evaluates the probability density function (PDF) at a selected point x = (x1, x2, x3). It also calculates the mean composition and, when all concentration parameters exceed 1, the interior mode. The PDF compares how strongly the chosen parameters support one location on the simplex relative to nearby locations; the mean and mode summarize the distribution's shape in different ways.
For this continuous Dirichlet model, a PDF is not the probability of one exact composition. An exact-point probability is zero. Instead, density describes relative concentration around that point: high density means nearby compositions are comparatively favored, while low density identifies a less favored region of the simplex.
How to read Dirichlet α and composition inputs
The three α inputs, α1, α2, and α3, are strictly positive Dirichlet concentration parameters. Their relative sizes determine the average composition. A larger α1 relative to α2 and α3 shifts the distribution toward larger x1 values. Equal α values make the distribution symmetric, although symmetry can still favor the center or the corners depending on the common value.
For a three-variable Dirichlet distribution, α0 = α1 + α2 + α3 is the total concentration. Larger totals cluster compositions more tightly around the mean. Values below 1 tend to place density near simplex edges and corners, where one or two components dominate. When every α value is above 1, density is pulled inward and the distribution has an interior mode; otherwise its maximum is on the boundary.
The x1, x2, and x3 fields define the composition at which this calculator evaluates density. They are unitless proportions, not counts, and must be positive and sum to 1. Enter 0.2 for 20% and 0.35 for 35%. Convert percentages by dividing by 100; convert counts to shares first. For example, counts 20, 30, and 50 correspond to 0.2, 0.3, and 0.5.
This implementation evaluates only interior simplex points, so each entered x value must be greater than 0. Dirichlet boundary behavior can require limits, especially for α values below 1 where density may diverge near an edge. Restricting entries to positive components keeps the calculator's evaluation rule unambiguous.
The Dirichlet PDF, mean, and mode formulas
For the three-component Dirichlet distribution calculated on this page, the PDF is
Here α0 = α1 + α2 + α3. The gamma function Γ extends factorials to non-integer parameter values. The calculator uses this normalization coefficient and the three component power terms directly to produce its displayed density.
For this Dirichlet model, each component's mean is
Thus the mean is the α vector divided by its total. Multiplying every α by the same factor leaves that mean unchanged but increases concentration. If all three α values exceed 1, the calculator also reports the interior mode:
The density depends on how the entered composition aligns with the α-shaped simplex distribution. In particular, the same composition can have very different PDF values under different concentration totals or different balances among α1, α2, and α3.
Worked example: evaluating a three-part Dirichlet density
Consider the composition (x1, x2, x3) = (0.2, 0.3, 0.5) with α = (2, 3, 5). This example places the evaluation point at the mean, while also showing why the mean and mode need not coincide. The total concentration is:
α0 = 2 + 3 + 5 = 10
The gamma-based coefficient is:
Γ(10) / [Γ(2)Γ(3)Γ(5)] = 362880 / 48 = 7560
The three Dirichlet power terms are:
0.21 × 0.32 × 0.54 = 0.2 × 0.09 × 0.0625 = 0.001125
Multiplying the coefficient and power terms gives:
PDF = 7560 × 0.001125 = 8.505
The calculator's mean output is:
mean = (0.2000, 0.3000, 0.5000)
Because every α value exceeds 1, the interior mode exists:
mode = ((2 - 1) / 7, (3 - 1) / 7, (5 - 1) / 7) = (0.1429, 0.2857, 0.5714)
This Dirichlet example distinguishes two summaries. The mean is the expected composition, whereas the mode is the interior location of greatest density. With unequal α values, subtracting 1 in the mode formula can shift the mode more strongly toward the component with the largest α.
How to interpret the Dirichlet result panel
After selecting Evaluate, the Dirichlet result panel lists the PDF, the mean, and the mode when the mode is defined. Check the simplex constraint before interpreting the result: all α values and all x values must be positive, and the three x values must sum to 1. A rounded set of shares that does not total 1 is not a valid evaluation point for this calculator.
Interpret the PDF comparatively. A larger value means the entered composition is in a relatively more favored region under the selected Dirichlet parameters; it does not mean a probability greater than 100%, and a density above 1 is entirely possible. Comparing nearby simplex points or changing α values reveals whether the distribution is sharply concentrated or diffuse.
The mean is usually the clearest share summary, especially for a Bayesian prior or posterior. The mode identifies the interior peak only when it exists. If this calculator omits the mode, at least one α is not greater than 1, so the distribution's highest density is associated with a boundary direction rather than a single interior composition.
Common three-variable Dirichlet parameter patterns
These Dirichlet parameter sets show how α changes the simplex shape before you evaluate any particular composition point.
| Parameters α |
Mean |
Interior mode? |
What the shape looks like |
| (1, 1, 1) |
(0.3333, 0.3333, 0.3333) |
No unique interior mode |
Uniform over the simplex; no composition is preferred over another. |
| (4, 4, 4) |
(0.3333, 0.3333, 0.3333) |
Yes, at the center |
Symmetric and strongly concentrated around balanced mixtures. |
| (0.5, 0.5, 0.5) |
(0.3333, 0.3333, 0.3333) |
No |
Mass is pushed toward corners and edges, so extreme compositions dominate. |
| (6, 2, 2) |
(0.6000, 0.2000, 0.2000) |
Yes, at (0.7143, 0.1429, 0.1429) |
Biased toward large x1, with moderate concentration away from the edges. |
In Bayesian uses, α values are often given pseudo-count intuition. The symmetric setting α = (1, 1, 1) is flat over three-part compositions. Larger equal values express more concentration around equal shares, while unequal values such as (8, 2, 1) express a prior tendency for the first component to be largest.
Dirichlet calculator assumptions and practical limits
This Dirichlet distribution page is limited to three variables, positive α parameters, and a strictly interior composition point. It does not integrate probability over custom simplex regions, estimate α from observations, or evaluate higher-dimensional Dirichlet distributions. Those tasks need different numerical methods and interfaces.
Dirichlet density calculations can also be numerically sensitive. Very large α values and compositions extremely close to a boundary may yield values spanning many orders of magnitude. The calculator applies the displayed formula, but output rounding can conceal small distinctions. Research workflows may require log densities, explicit precision control, and stated boundary conventions in a statistics package.
Within those limits, this calculator provides a direct check of whether a proposed three-part composition is relatively favored by chosen α parameters. Its PDF, mean, and conditional interior-mode outputs are useful for teaching simplex geometry, checking Bayesian priors, and comparing plausible concentration settings.