Dirichlet-Distributed Outputs

• Dirichlet-distributed outputs are probability vectors following the Dirichlet law that model compositional data in areas such as random matrix products, Bayesian inference, and stochastic processes.
• They exhibit sparsity in high-dimensional regimes, enabling efficient topic modeling, clustering, and variational inference by focusing on dominant coordinates.
• Advanced frameworks include stochastic diffusions, generalized and non-central extensions, and privacy-preserving mechanisms, offering robust tools for modern statistical inference.

Dirichlet-distributed outputs are probability vectors (or measures) whose joint distribution follows the Dirichlet law, often arising as stationary, limiting, or posterior distributions in stochastic processes, Bayesian modeling, random matrix theory, and number theory. The Dirichlet family—including its generalizations—supports rich modeling of compositional, exchangeable, and often constrained data, due to its inherent simplex geometry and versatile parameterization. This article summarizes the principal mathematical frameworks in which Dirichlet-distributed outputs appear, including convergence of random stochastic matrix products, sparse outputs for small parameter regimes, stochastic diffusions, Bayesian regression and mixture modeling, generalized and non-central extensions, as well as applications in random combinatorial structures and privacy-preserving data analysis.

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1. Convergence to Dirichlet Distributed Limits in Random Matrix and Exchange Models

A cornerstone result in the probabilistic theory of Dirichlet-distributed outputs is the asymptotic appearance of the Dirichlet law as the limit of infinite products of random stochastic matrices. Let $X$ be a random $d \times d$ stochastic matrix and $\{X(n)\}_{n \geq 1}$ i.i.d. copies, with the "left product" defined by

$X(n,1) = X(n) X(n-1) \cdots X(1).$

Chamayou and Letac (1994) originally proved that under the condition that the rows of $X$ are independent Dirichlet vectors (with matching row sums of parameters), the limit $\lim_{n \to \infty} X(n,1)$ exists almost surely as a matrix with identical Dirichlet-distributed rows.

The extension in (McKinlay, 2012) greatly relaxes these constraints, showing that two conditions are necessary and sufficient for almost sure convergence to Dirichlet-distributed outputs:

• [C1] There exists a vector $t \in \mathbb{R}_+^d$ such that, for a Gamma vector $V \sim G(t)$ (components independent Gamma variables), $V X \overset{d}{=} V$.
• [C2] There exists $m \geq 1$ with $P[X(m,1) \text{ is positive}] > 0$.

When these are satisfied, the limiting matrix has identical Dirichlet-distributed rows with parameters $t$:

$X(1) \sim \text{Dirichlet}(t).$

Further, the Dirichlet law is stationary under right-multiplication by $X$:

$$Y \sim \text{Dirichlet}(t),\ Y \perp X \implies YX \overset{d}{=} Y.$$

Applications: These conditions encompass a wide array of applied models:

• Random exchange models (e.g., distributed averaging, wealth exchange): The stationary distribution of agent proportions is Dirichlet.
• Random nested simplices: The limiting barycentric coordinates converge to a Dirichlet law.
• Service networks with polling: Cyclically updated systems with random mixing admit Dirichlet stationary measures.

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2. Sparsity and High-Dimensional Regimes

Dirichlet draws with parameters $\alpha < 1$ exhibit sparsity: most coordinates are near zero, while a few dominate. The precise result in (Telgarsky, 2013) states that for $(X_1,\ldots, X_n) \sim \text{Dirichlet}(\alpha)$, for any $\epsilon > 0$ and $k \in \mathbb{N}$,

$$\Pr\left[\#\{i : X_i \geq \epsilon\} \leq k\right] \geq 1 - \epsilon^{-n\alpha} e^{-(k+1)/3} - e^{-4(k+1)/9}.$$

For $\alpha = 1/n$ and $\epsilon = 1/n^{c_0}$, most coordinates fall below $1/n^{c_0}$, with the number of large coordinates scaling only as $\mathcal{O}(\log n)$. This sparsity is exploited in:

• Topic modeling (latent Dirichlet allocation): Sparse topic distributions for documents.
• Clustering: Strong cluster assignments in mixtures.
• Efficient variational inference: Computation focuses on non-negligible coordinates.

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3. Stochastic Diffusions with Dirichlet Invariant Measures

A methodology for constructing Itô diffusions with Dirichlet stationary distributions is presented in (Bakosi et al., 2013). For $N$ variables $Y$ (components summing to 1), the SDEs are:

$$dY_\alpha = \frac{b_\alpha}{2}[S_\alpha Y_N - (1 - S_\alpha) Y_\alpha]\,dt + \sqrt{\kappa_\alpha Y_\alpha Y_N}\,dW_\alpha,$$

with $Y_N = 1 - \sum_{\beta=1}^{N-1} Y_\beta$. The parameters are related to the Dirichlet shape parameters $\omega_\alpha$ via:

$$\omega_\alpha = \frac{b_\alpha}{\kappa_\alpha} S_\alpha,\quad \omega_N = \frac{b_\alpha}{\kappa_\alpha}(1 - S_\alpha).$$

The diffusion vanishes at the simplex boundaries (where $Y_\alpha=0$ or $Y_N=0$), ensuring the unit-sum constraint is preserved. Closely related is the multivariate Wright–Fisher (WF) diffusion, which employs a full rather than diagonal diffusion matrix but also has Dirichlet invariant law.

This framework generalizes to the Lochner generalized Dirichlet distribution (Bakosi et al., 2013):

• Additional covariance structure and drift terms are introduced, yielding a diffusion for $\mathcal{G}(Y, \alpha, \beta)$, enhancing flexibility (including positive covariances) compared to the standard Dirichlet.

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4. Dirichlet Mixture Models and Infinite-Dimensional Processes

Bayesian nonparametric models frequently produce Dirichlet-distributed outputs, as in Dirichlet process mixture models (DPMMs). In DPMMs, the mixing proportions $$(\pi_1, \ldots, \pi_K, \pi_{\textrm{new}}) \sim \text{Dir}(N_1, \ldots, N_K, \alpha)$$, with $N_k$ the number assigned to cluster $k$ and $\alpha$ the concentration parameter. These finite-D draws approximate Dirichlet processes $DP(\alpha, G_0)$ via stick-breaking constructions in the infinite-dimensional limit (Wang et al., 2017, Dinari et al., 2022).

Recent work has addressed distributed scalable inference for DPMMs in high performance environments:

• Asynchronous, distributed sampling: Workers independently create components, later merged via probabilistic consolidation schemes to ensure global statistical consistency while minimizing inter-node communication.
• GPU-based and CPU-based implementations (Dinari et al., 2022): Leverage parallelism to scale inference to very large datasets, with Dirichlet-distributed outputs appearing in both cluster weights and sub-cluster structures.

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5. Generalizations: Conditional Non-Central, Rank-Ordered, and Generator-Based Extensions

Conditional Non-Central Dirichlet Distributions (Orsi, 2021) introduce non-centrality parameters to flexibly adjust the behavior near simplex vertices. The density can be expressed as a mixture of Dirichlet laws:

$$\text{CNcDir}^D(x; \alpha, \lambda) = \sum_{j \in \mathbb{N}_0^{D+1}} Pr(N=j)\, \text{Dir}^D(x; \alpha+j),$$

and equivalently as a "perturbed" Dirichlet density by products of confluent hypergeometric functions. Arbitrary positive finite limits at vertices (corners) can be attained by tuning non-centrality.

Generalized Rank Dirichlet Distributions (GRD) (Itkin, 2023) extend the Dirichlet to the ordered simplex (decreasing order) and allow negative parameters, subject to tail sum constraints. This class models ranked weight vectors, invariant distributions for ranked capital in financial mathematics, and supplies closed-form methods for moments and simulation algorithms.

Dirichlet Generator Mechanism (Arashi et al., 2019) generalizes beta-generated distributions to the multivariate case, embedding baseline CDFs (e.g., Gamma, Pareto) within the Dirichlet kernel:

$$h(x_1,\ldots,x_p) = \frac{1}{B(\alpha)} (1 - \sum_{i=1}^p G_i(x_i))^{\alpha_{p+1}-1} \prod_{i=1}^p g_i(x_i) G_i(x_i)^{\alpha_i-1}.$$

This construction maintains Dirichlet marginal structure and is particularly suitable for compositional data.

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6. Dirichlet-distributed Outputs in Decision Theory, Privacy, and Combinatorics

Exceedance Probabilities (EPs) (Soch et al., 2016) for Dirichlet random vectors quantify the probability that a particular component is maximal:

$\phi_j = P(r_j = \max[r]).$

Computationally, EPs are efficiently evaluated via a numerical integration approach involving the gamma representation of the Dirichlet.

Private KL minimization via the Dirichlet mechanism (Ponnoprat, 2021): By casting the exponential mechanism with KL loss, privatized empirical distributions are released as Dirichlet-distributed samples, with parameters tuned to guarantee differential (Rényi) privacy. This mechanism outperforms additive noise mechanisms in tasks such as private maximum likelihood and classification due to its alignment with the KL-based utility objective.

Factorizations and Number Theory (Leung, 2022): The vector of normalized logarithms of integer factorizations into $k$ parts converges in distribution to $\text{Dir}(1/k,\ldots,1/k)$. Mellin transform and multidimensional complex analytic techniques provide precise rates and allow for Dirichlet limit laws with general parameters through weighted factorizations.

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7. Statistical Inference and Bayesian Regression for Dirichlet-Distributed Data

Dirichlet-distributed outputs are standard for modeling vector-valued proportions or compositions in Bayesian statistics. The typical regression scenario (Sennhenn-Reulen, 2018) models observations $Y_i$ on the simplex as Dirichlet responses parameterized by a mean vector $\mu_i$ (expressed via a softmax-linear link function of covariates) and a precision parameter $\theta$:

$Y_i \sim \text{Dirichlet}(\mu_i \cdot \theta).$

Priors on coefficients, careful identifiability handling (e.g., reference level constraints), and computational implementations (notably in Stan with HMC sampling) yield full posterior inference and uncertainty quantification for compositional regression. The method accommodates changing compositions as functions of covariates, properly quantifies uncertainty across all components (even reference categories), and avoids the need for ad hoc normalization post-processing.

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References (arXiv id in brackets)

• Random stochastic matrices and Dirichlet limits: (McKinlay, 2012)
• Sparsity of Dirichlet draws: (Telgarsky, 2013)
• Stochastic diffusions and Dirichlet/GDir processes: (Bakosi et al., 2013, Bakosi et al., 2013)
• Generalized Dirichlet random walks: (Gregorio, 2013, Letac et al., 2013, Caer, 2015)
• Exceedance probabilities and applications: (Soch et al., 2016)
• Distributed DPMM estimation, scalable inference: (Wang et al., 2017, Dinari et al., 2022)
• Dirichlet process and iterated Dirichlet processes: (Donald et al., 1 May 2025)
• Dirichlet-generated (DG) distributions: (Arashi et al., 2019)
• Conditional non-central Dirichlet: (Orsi, 2021)
• Rank Dirichlet distributions: (Itkin, 2023)
• Number-theoretic applications: (Leung, 2022, Inoue et al., 2022)
• Privacy-preserving mechanisms: (Ponnoprat, 2021)
• Bayesian Dirichlet regression: (Sennhenn-Reulen, 2018)
• Dirichlet series and records: (Peled et al., 2015)