Dirichlet-Distributed Continuous Variables

• Dirichlet-distributed continuous variables are real-valued vectors on the simplex where each component is non-negative and sums to one.
• They serve as a foundational tool in Bayesian statistics, population genetics, and compositional data analysis due to their conjugacy and tractability.
• Their elegant properties, including neutrality and a sharp Poincaré inequality, underpin diffusion models with exponential convergence in both finite and infinite dimensions.

Dirichlet-distributed continuous variables are real-valued vectors constrained to the simplex: each component is non-negative and all components sum to one. The Dirichlet distribution, parameterized by positive real parameters, provides a probability law for such vectors. It is a foundational object in probability, Bayesian statistics, machine learning, population genetics, and compositional data analysis because it encodes distributions over proportions and compositional structures. Owing to its elegant mathematical properties—conjugacy, closure under aggregation, neutrality, and tractability—the Dirichlet distribution underpins both classical and modern stochastic modeling and inference involving continuous variables constrained to the simplex.

1. Mathematical Structure and Key Properties

The Dirichlet distribution with parameters $$\alpha = (\alpha_1, \ldots, \alpha_{N+1}) \in (0,\infty)^{N+1}$$ is defined on the $N$-simplex: $$\Delta^{(N)} := \left\{ x \in [0,1]^N : \sum_{i=1}^N x_i < 1 \right\}.$$ Its density is: $$\frac{\Gamma\left(\sum_{i=1}^{N+1} \alpha_i\right)}{ \prod_{i=1}^{N+1} \Gamma(\alpha_i)} x_1^{\alpha_1 - 1} \cdots x_N^{\alpha_N - 1} (1 - \sum_{i=1}^N x_i )^{\alpha_{N+1} - 1}, \qquad x \in \Delta^{(N)}.$$ Key mathematical properties include:

• Aggregation and partition: If one sums a subset of components, the resulting marginal distribution remains Dirichlet, with parameters equal to the sums over the partitioned indices.
• Neutrality: By a change of coordinates, a Dirichlet vector can be represented via a sequence of independent Beta-distributed variables.
• Pólya urn scheme: Dirichlet arises as the limit of proportions in generalized Pólya urn processes.
• Conjugacy: In Bayesian statistics, the Dirichlet is conjugate to the multinomial, making posterior computations tractable.
• Applications: It appears naturally in allele frequency models in population genetics.

2. Spectral Theory, Poincaré Inequality, and Associated Diffusions

A central analytical feature of the Dirichlet distribution is its role as the stationary measure for certain diffusion processes on the simplex. Consider the Wright-Fisher-type stochastic differential equation: $$dX_i(t) = \left[ \alpha_i \big(1 - |X(t)|_1\big) - \alpha_{N+1} X_i(t) \right] dt + \sqrt{2 (1 - |X(t)|_1) X_i(t)}\, dB_i(t),$$ for $1 \le i \le N$, where $B_i(t)$ are independent Brownian motions. This process is reversible with respect to the Dirichlet law and ensures that the sample remains within the simplex at all times.

An explicit Dirichlet form is associated to the process: $$\mathcal{E}_\alpha^{(N)}(f, f) = \int_{\Delta^{(N)}} (1 - |x|_1) \sum_{i=1}^N x_i (\partial_i f(x))^2 \, \mu_\alpha^{(N)}(dx).$$ The Poincaré inequality for Dirichlet distributions is: $$\mu_\alpha^{(N)}(f^2) \leq \frac{1}{\alpha_{N+1}} \mathcal{E}_\alpha^{(N)}(f, f), \quad \mu_\alpha^{(N)}(f) = 0,$$ and the constant $1/\alpha_{N+1}$ is sharp, being the inverse spectral gap of the associated generator. As a result, the associated diffusion process converges exponentially fast in $L^2$ to the Dirichlet invariant law, at rate $\alpha_{N+1}$.

In the infinite-dimensional setting, let

$$\Delta^{(\infty)} = \left\{ x \in [0,1]^\mathbb{N} : \sum_{i=1}^\infty x_i \le 1 \right\}$$

and the analogous Poincaré inequality persists, with spectral gap $\alpha_0$ if $\alpha = (\alpha_i)_{i\ge 1}$, $\alpha_0 > 0$, and $\sum \alpha_i < \infty$.

For discrete population models (with immigration and emigration), the spectral gap of the transition chain's generator matches the diffusive continuous limit, again equal to $\alpha_{N+1}$. Thus, both finite and infinite-dimensional Dirichlet structures possess an explicit and optimal spectral theory.

3. Functional Inequalities and Ergodic Properties

The explicit Poincaré inequality for Dirichlet distributions enables sharp control of variances of functions under the measure, characterizing the speed of ergodic convergence of processes towards equilibrium. Homogeneous polynomials of degree $d$ in $N$ variables form invariant subspaces for the generator $L_\alpha^{(N)}$, allowing recursive explicit computation of the generator's full spectrum. The lowest nonzero eigenvalue (i.e., the spectral gap) is $\alpha_{N+1}$, and all higher eigenvalues are obtained through a combinatorial recursion involving the degrees of polynomials and parameter sums. This spectral information controls rates of convergence and stability in models built upon Dirichlet-distributed continuous variables.

4. Scaling Limits and Connections to Infinite Dimensions

The Dirichlet distribution’s construction naturally extends to infinite-dimensional spaces as limits of finite-dimensional distributions. Specifically, for suitable $(\alpha_i)_{i\ge1}$ with $\sum \alpha_i<\infty$, the Dirichlet process on $\Delta^{(\infty)}$ is the weak limit of finite Dirichlet distributions on $\Delta^{(N)}$ as $N\to\infty$, preserving properties such as conjugacy, neutrality, and aggregation over partitions. Analytical tools (such as Dirichlet forms and spectral theory) carry over, with Poincaré inequalities and spectral gap results holding in the limit.

5. Applications and Modeling Implications

Dirichlet-distributed continuous variables are fundamental in hierarchical Bayesian inference, population genetics, and compositional data analysis:

• Bayesian inference: Conjugate to multinomial likelihoods, Dirichlet priors enable tractable computation of posterior distributions over categorical probabilities.
• Stochastic modeling: Dirichlet distributions describe stationary distributions of diffusive processes constrained by conservation laws (e.g., proportions summing to one) and are instrumental in modeling ecological, genetic, and chemical systems.
• Compositional data: The Dirichlet's simplex constraint makes it suitable for applications where responses are inherently relative or compositional (e.g., market shares, biomarker fractions).

Knowledge of sharp spectral gaps and explicit functional inequalities enables precise control over the mixing rates and convergence behavior in simulation or inference algorithms based on Dirichlet models.

6. Summary Table

Topic	Formula / Statement	Key Point
Dirichlet distribution density	$$\propto \prod_{i=1}^N x_i^{\alpha_i-1}(1-|x|_1)^{\alpha_{N+1}-1}$$	On $\Delta^{(N)}$, params $\alpha$
Poincaré inequality	$$\mu(f^2) \leq \frac{1}{\alpha_{N+1}} \mathcal{E}(f,f),\; \mu(f) = 0$$	Constant is sharp
Diffusion generator	$$L_\alpha^{(N)} f = \sum_{i=1}^N [(1-|x|_1)x_i \partial_i^2 f + (\alpha_i(1-|x|_1)-\alpha_{N+1}x_i)\partial_i f]$$	Wright-Fisher type process
Convergence rate	$$\|P_t f - \mu(f)\|_{L^2} \leq e^{-\alpha_{N+1} t} \|f-\mu(f)\|_{L^2}$$	Exponential convergence, rate = spectral gap
Infinite Poincaré inequality	$\mu(f^2) \leq \frac{1}{\alpha_0} \mathcal{E}(f,f)$ for the Dirichlet process	Spectral gap $\alpha_0$
Discrete model spectral gap	Spectral gap $= \alpha_{N+1}$ for finite Markov chain	Consistent with diffusion case
Spectrum of generator	Recursive: $$\mathscr{A}_{d+1} = (2d+|\alpha|_1+\mathscr{A}_d)\cup\{0^{C(N,d+1)-C(N,d)}\}$$	Full spectrum explicitly characterized

7. Conclusion

The Dirichlet distribution's analytic, probabilistic, and ergodic properties are precisely characterized by sharp functional inequalities, explicit spectral gap formulas, and convergence theory for associated diffusion processes. These results scale naturally to infinite-dimensional settings and to discrete population models, providing a robust foundation for modeling, analysis, and computation involving Dirichlet-distributed continuous variables across applied mathematics, statistics, and the sciences.