Multinomial Thinning Decomposition

• Multinomial thinning decomposition is a collection of probabilistic methods that partition and represent discrete data through thinning operations, serving as a discrete analog of continuous scaling.
• This approach facilitates Poisson approximation, multifractal analysis, and efficient computation via thinning Markov chains, Poisson–Charlier expansions, and stick-breaking methods.
• Its applications span statistical physics, combinatorics, computational statistics, and data analysis, offering actionable insights for modeling complex discrete systems.

Multinomial Thinning Decomposition is a collection of probabilistic and information-theoretic techniques for partitioning, representing, and analyzing discrete random variables, random vectors, and processes using thinning operations that generalize scaling in the discrete domain. Thinning replaces continuous scaling by the probabilistic retention or sub-allocation of counts, a concept central to modern developments in Poisson approximation, multifractal process theory, computational statistics, and information decomposition. Multinomial thinning has structural and computational implications for statistical physics, combinatorics, data analysis, and the paper of limit theorems for random discrete objects.

1. Fundamental Principles: Definition and Mechanisms

Multinomial thinning is defined for a discrete probability distribution $P$ on $\mathbb{N}_0$ or a multinomial vector. For fixed $\alpha \in [0,1]$, the $\alpha$-thinned distribution $T_{(\alpha)}(P)$ is constructed by independently "keeping" each unit of a count $X\sim P$ with probability $\alpha$, leading to the mapping: $$T_{(\alpha)}(P)(z) = \sum_{x=z}^{\infty} P(x) \binom{x}{z} \alpha^z (1-\alpha)^{x-z}$$ This formula defines a discrete analog of continuous scaling (0906.0690). For vector-valued or multinomial random variables, thinning generalizes to the splitting of total counts into subchannels according to multinomial probabilities, and to thinning invariant transformations for more complex partition structures (Starr et al., 2011).

Thinning extends naturally to point processes, compound distributions, and to joint random objects (e.g., order statistics, record processes), with thinning operators serving as the foundational mechanism for discrete analogs of central limit theorems, compound Poisson constructions, and multifractal scaling (Aldridge, 20 Feb 2025, Grahovac, 26 Sep 2025).

2. Law of Thin Numbers and Poisson Approximation

The law of thin numbers establishes that under appropriate thinning and convolution operations, discrete distributions converge to Poisson laws, closely paralleling the classical central limit theorem for sums of continuous random variables. For i.i.d. $X_1, X_2, \dots$ with mean $\lambda$, the thinned convolution

$$\lim_{n\to\infty} D(T_{(1/n)}(P^{*n}) \Vert \mathrm{Po}(\lambda)) = 0$$

holds for relative entropy $D(\cdot\Vert\cdot)$ under mild conditions (0906.0690).

This convergence admits precise rates and nonasymptotic bounds. For ultra-log-concave $P$,

$$D(T_{(1/n)}(P^{*n}) \Vert \mathrm{Po}(\lambda)) \leq \frac{C}{n^2}$$

for some constant $C$. Convergence holds in relative entropy, total variation, and entropy senses, and extends to compound thinning (where particles retained by thinning are replaced by draws from a fixed compounding law, yielding compound Poisson limits).

Analogous phenomena occur for point processes: thinning (with fixed probability $1/n$) after superposition of $n$ IID processes yields weak convergence to a Poisson process (Aldridge, 20 Feb 2025).

3. Thinning Markov Chains and Analogy with Ornstein–Uhlenbeck

A family of operators, $$U_{(\alpha)}^{\lambda}(P) = T_{(\alpha)}(P) * \mathrm{Po}((1-\alpha)\lambda)$$, forms a Markov semigroup with the composition property

$$U_{(\alpha)}^{\lambda}\circ U_{(\beta)}^{\lambda} = U_{(\alpha\beta)}^{\lambda}$$

Mapping $\alpha = \exp(-t)$ yields a continuous-time semigroup with unique invariant measure $\mathrm{Po}(\lambda)$ (0906.0690). This thinning Markov chain is strongly analogous to the Ornstein–Uhlenbeck process in Gaussian settings, serving as the discrete mechanism underlying the entropy power inequality for Poisson approximation.

The approach leverages orthonormal polynomial expansions (Poisson–Charlier system) to capture rates of convergence: for the first nonzero Poisson–Charlier moment $E[P_\kappa^\lambda(X)]$,

$$\chi^2(U_{(\alpha)}^\lambda P, \mathrm{Po}(\lambda)) \sim \alpha^{2\kappa} E[P_\kappa^\lambda(X)]^2 \quad \text{as } \alpha\downarrow 0$$

with information divergence tightly controlled by the $\chi^2$-distance.

4. Generalizations: Partition Structures, Compound Thinning, and Data Decomposition

Multinomial thinning generalizes beyond i.i.d. cases and basic sequences to partition structures, gap configurations, multivariate processes, and compound discrete objects (Starr et al., 2011, Dharamshi et al., 2023, Rooij, 12 Feb 2024). For such cases, thinning may preserve distributional invariance, giving rise to thinning invariant measures (notably Poisson–Kingman and Poisson–Dirichlet distributions for partition structures). Thinning invariant laws are rigorously characterized for sequences and conjectured (using Cox construction and Poissonization) for gaps and partitions.

Generalized data thinning expands the methodology using sufficient statistics: instead of summation, any deterministic function $T$ such that $X = T(X^{(1)},..., X^{(K)})$ and $T$ is sufficient for the parameter can be employed (Dharamshi et al., 2023). This provides a bridge between sample splitting, convolution-based thinning, and more general information-preserving decompositions, enabling applications in model validation, post-selection inference, and robust sample partitioning.

In complex models (e.g., multinomial regression for multivariate binary data), canonical thinning decomposition allows the parameter vector to be factored as $m_k + u_i'v_k$, with $u_i$ and $v_k$ constrained by external variables, providing direct interpretational links between predictors, profiles, and log odds/odds ratios (Rooij, 12 Feb 2024).

5. Mathematical and Computational Aspects

Thinning decompositions are not only conceptual but also yield tractable representations, step-function quantile mappings, and computational algorithms. For triangular arrays approximating Gaussian limits by multinomial sums, admissible permutations encode the transition between level sets and quantile functions in a manner that is polynomial-time computable. These representations are constructed so each component depends only on a finite number of input bits (Dobric et al., 2016).

The stick-breaking construction, in combination with Pólya–Gamma augmentation, enables the multinomial likelihood to be rewritten as a product of conditionally Gaussian factors

$$p(x \mid \psi) = \prod_{k=1}^{K-1} \binom{N_k}{x_k} \sigma(\psi_k)^{x_k} [1-\sigma(\psi_k)]^{N_k-x_k}$$

where each binomial term is augmented by a Pólya–Gamma variable, so that efficient Bayesian inference via block Gibbs or variational methods becomes possible. The dependencies between categories and the potential for complex structured prior modeling are far superior to standard Dirichlet–multinomial approaches (Linderman et al., 2015).

6. Extensions, Applications, and Connections

Thinning and multinomial thinning decomposition underpin diverse research themes and applications:

• Partial Information Decomposition: Multinomial thinning is central to information-theoretic decompositions of mutual information into unique, redundant, and synergistic terms in PID, enabling closed-form calculations when specific Markov chain conditions are met (Goswami et al., 2023).
• Integer-valued Multifractal Processes: Thinning operations are extended to integer-valued processes for multifractal scaling, with thinning replacing continuous multiplication and moment scaling characterized via nonlinear structure functions and cascade clocks (Grahovac, 26 Sep 2025).
• Refinements and Order Statistics: The joint distribution of multinomial outcome counts and permutation statistics (e.g., inversion counts) yields refined representations informing on order/homogeneity of outcomes, relevant in diagnostics and conditioning (Sills, 9 Sep 2024).
• Invariant Partition Structures and Statistical Physics: The role of thinning decomposition in equilibrium laws for partition structures connects to spin glass theory via entropy shift and Poisson–Dirichlet fixed points (Starr et al., 2011).

In recommendation systems, preference learning, and social choice contexts, mixtures of multinomial logit models can be statistically and computationally learned from sparse ordinal data via tensor decomposition plus spectral ranking, contingent on precise incoherence and graph-theoretic conditions (Oh et al., 2014).

7. Key Formulas and Technical Features

Operation/Quantity	Formula/Construction	Domain
Thinned distribution (Renyi)	$$T_{(\alpha)}(P)(z) = \sum_x P(x) \binom{x}{z} \alpha^z (1-\alpha)^{x-z}$$	Discrete RV ($\mathbb{N}_0$)
Thinning Markov chain	$$U_{(\alpha)}^\lambda(P) = T_{(\alpha)}(P) * \mathrm{Po}((1-\alpha)\lambda)$$	Law of Thin Numbers
Stick-breaking/Pólya-Gamma augm.	$$p(x, \omega | \psi) \propto \prod_k \exp((x_k-N_k/2)\psi_k - \omega_k \psi_k^2/2)$$	Bayesian inference
Poisson–Charlier expansion	$$\chi^2(U_{(\alpha)}^\lambda P, \mathrm{Po}(\lambda)) \sim \alpha^{2\kappa} E[P_\kappa^\lambda(X)]^2$$	Rates of convergence
Multinomial canonical decomposition	$\theta_{ik} = m_k + u_i' v_k$, $U = X B_x$, $V = Z B_z$, $m = W b_w$	Categorical regression
Refined multinomial with inversions	$$P(Y = y, I = i) = \operatorname{inv}(y_1,\dots,y_k; i)\prod_j p_j^{y_j}$$	Joint order/count distribution

Multinomial thinning decomposition, and its generalizations, provide a technically robust and flexible toolkit for generating Poisson approximations, constructing efficient representations for multinomial data, quantifying dependencies and information flow, and modeling discrete stochastic systems where thinning, invariance, and decomposition play central roles.