Wallenius Noncentral Hypergeometric Distribution

• The Wallenius noncentral hypergeometric distribution is a probabilistic model for sampling without replacement from weighted, partitioned populations, generalizing the standard hypergeometric case.
• It is applied in biased urn models, contingency table analysis, and randomized scheduling, offering a framework for modeling selection bias in various practical settings.
• Computation relies on numerical integration and optimization methods, with both likelihood-based and Bayesian inference techniques addressing its intractable normalizing constant.

The Wallenius noncentral hypergeometric distribution is a probabilistic model for sampling without replacement from a finite population partitioned into multiple categories, where each item has an associated category-specific weight or bias parameter. This model generalizes the standard hypergeometric distribution by permitting non-uniform selection probabilities and arises in statistical inference, randomized allocation, and preference modeling. The distribution is widely recognized for its foundational role in biased urn models, contingency table inference, and emerging applications in stochastic scheduling.

1. Mathematical Definition and Formulation

Let the population consist of $N=\sum_{i=1}^c m_i$ objects, partitioned into $c$ classes, with $m_i$ objects of class $i$. Assign to each class $i$ a strictly positive weight $\omega_i > 0$. Consider sampling $n$ objects sequentially without replacement, where at each step the probability of choosing an object from class $i$, given the current sample composition $\boldsymbol X_t = (X_{1t},\ldots,X_{ct})$, is given by

$$\Pr\{\,Z_{t+1}=i\mid \boldsymbol X_t\} = \frac{(m_i - X_{it}) \omega_i} {\sum_{j=1}^c (m_j - X_{jt}) \omega_j}.$$

The (joint) probability mass function (PMF) for the resulting count vector $\boldsymbol X = (X_1, \ldots, X_c)$, with $\sum_{i=1}^c X_i = n$, is given (Chesson, 1976) by

$$P(\boldsymbol x; n, \boldsymbol m, \boldsymbol \omega) = \left[\prod_{i=1}^c \binom{m_i}{x_i}\right] \int_0^1 \prod_{i=1}^c (1 - t^{\omega_i/d})^{x_i} \, dt,$$

with

$$d(\boldsymbol x, \boldsymbol \omega) = \sum_{i=1}^c \omega_i (m_i - x_i).$$

For the univariate (two-category) case, the PMF simplifies to

$$P(X=k) = \binom{M_1}{k}\binom{M_2}{n-k} \frac{ \int_0^1 t^{n-1} (1-t^{w_1})^{M_1-k}(1 - t^{w_2})^{M_2 - (n-k)}\,dt }{ \int_0^1 t^{n-1} (1-t^{w_1})^{M_1}(1-t^{w_2})^{M_2}\,dt },$$

with weights often parameterized as $w_1>0$ and $w_2=1$ for interpretability (Haines, 11 Nov 2025).

In the multivariate extension, the normalized PMF becomes: $$P(X_1=x_1,\dots,X_c=x_c\mid \mathbf{M},n,\mathbf{w}) = \left[\prod_{i=1}^c\binom{M_i}{x_i}\right] \frac{ \int_{0}^{1} t^{n-1} \prod_{i=1}^c (1 - t^{w_i})^{M_i - x_i} dt }{ \int_{0}^{1} t^{n-1} \prod_{i=1}^c (1 - t^{w_i})^{M_i} dt },$$ where $\sum x_i = n$ (Haines, 11 Nov 2025).

2. Properties, Special Cases, and Odds Ratio

If all weights are equal, i.e., $\omega_i = \omega$ for all $i$, the distribution reduces to the central (ordinary) multivariate hypergeometric law. The mean in this case is

$\mathbb{E}[X_i] = n \frac{m_i}{N},$

the same as in the classical model (Grazian et al., 2017).

In the univariate ($c=2$) case, defining the single noncentrality parameter $\theta = w_1 / w_2$, the odds of drawing a ball from category 1 versus category 2 in a single draw is exactly $\theta$ above the baseline $M_1/M_2$. Thus, for $w_2=1$,

$\text{Odds ratio} = \theta = w_1,$

establishing a direct connection to the classical odds ratio in $2 \times 2$ contingency analysis (Haines, 11 Nov 2025).

Closed-form expressions for the mean, variance, or higher moments under general weights are unavailable due to the integral representation of the PMF. Differentiation under the integral or numerical methods (e.g., as in Fog, 2008) are necessary (Grazian et al., 2017).

3. Computational and Inferential Methodology

Analytic computation of the PMF requires evaluation of an intractable (no closed-form) normalizing constant involving a multivariate integral. For moderate or large $c, n$, numerical instability and computational inefficiency are significant concerns.

Likelihood-Based Inference:

• The log-likelihood for observed counts $\mathbf{x}$ is

$$\ell(\mathbf{w}) = \sum_{i=1}^c \log \binom{M_i}{x_i} + \log I_1(\mathbf{w}) - \log I_0(\mathbf{w}),$$

where $I_1$ and $I_0$ are integrals over $[0,1]$, with different powers in the $(1-t^{w_i})$ terms.

• For optimization, weights are typically reparameterized via additive log-ratios or softmax transforms (mapping the simplex to $\mathbb{R}^{c-1}$), enabling unconstrained optimization (e.g., BFGS) (Haines, 11 Nov 2025).
• Numerical quadrature in log-space (using the log-sum-exp trick) is essential for computing the highly variable integrals and preventing underflow (Haines, 11 Nov 2025).

Bayesian Inference:

• Priors on the weights $\mathbf{w}$ are typically Dirichlet distributions over the simplex.
• The sphere-walk Metropolis (SWM) algorithm is used for MCMC sampling. This method proposes moves on the surface of the positive orthant of the $(c-1)$-sphere, mapping between Cartesian (sphere) coordinates $x_i = \sqrt{w_i}$ and barycentric (simplex) coordinates $w_i = x_i^2$, and applies Metropolis-Hastings acceptance criteria. No Jacobian correction is needed as it cancels in the ratio (Haines, 11 Nov 2025).

Approximate Bayesian Computation (ABC):

• For large $c$ and $n$, ABC rejection sampling is employed: posterior draws of $\boldsymbol\omega$ are accepted if simulated summary statistics closely match observed statistics under a total-variation metric, using only forward sampling from the Wallenius law (Grazian et al., 2017).

4. Applications and Illustrative Case Studies

Preference Modeling: The Wallenius distribution's ability to encode selection biases via weights allows modeling of individual or group preferences in ranked data. For example, movie ratings binned into genres are modeled as counts drawn from the Wallenius law, with posterior inference on $\omega_i$ revealing genre affinities. Posterior means and marginal distributions over weights are visualized via violin plots to quantify uncertainty (Grazian et al., 2017).

Contingency Table Analysis: In $2 \times 2$ and multigroup contingency tables, max-likelihood estimation and Bayesian inference for odds ratios are implemented using the univariate or multivariate Wallenius distribution, as demonstrated by modeling arm success in clinical trials or treatment effects in multi-arm experiments (Haines, 11 Nov 2025).

Randomized Scheduling and Stochastic Allocation: In scheduling systems (e.g., periodic multi-source systems with distinct Age of Information requirements), the probability of specific transmission events is governed by the multivariate Wallenius law. Analytical upper bounds for peak age-of-information (PAoI) violation probabilities are obtained through Chernoff bounds over the PMF, and numerical or analytic solutions for policy weights are derived (Lin et al., 29 Jan 2025).

5. Asymptotics, Approximations, and Computational Strategies

Given the lack of a closed-form for the PMF in most cases, asymptotic analysis and numerical methods are essential:

• Laplace-Stirling Approximations: For large $n$, the PMF exhibits a Laplace-type form,

$$g(y_{i,n-1}, n, I_n, p) \approx \exp\left\{ -n \Phi(T, y_{i, n-1}) \right\} \left[ 2\pi n \det \nabla^2 \Phi(T, y_{i, n-1}) \right]^{-1/2},$$

where $\Phi$ is a convex function defined on the configuration, and $\nabla^2\Phi$ its Hessian (Lin et al., 29 Jan 2025).

• Chernoff Bounds and Policy Design: Analytical bounds for rare-event probabilities (such as PAoI violations) exploit convexity and monotonicity properties of the PMF, with efficient parameter search strategies informed by these results (Lin et al., 29 Jan 2025).
• Numerical Quadrature and Scaling: Integrals in the Wallenius PMF are computed using log-scale quadrature, with scaling to prevent numerical underflow, essential for both likelihood evaluation and MCMC acceptance ratios (Haines, 11 Nov 2025).

6. Limitations, Numerical Subtleties, and Advanced Inference

Key computational challenges arise from the intractable normalizing integrals and sharp scaling of probabilities:

• Likelihoods can reach extremely small values; scaling by arbitrary factors (without impacting inference) is necessary in optimization routines (Haines, 11 Nov 2025).
• Transformations to unconstrained spaces (e.g., via log-ratio) simplify optimization but require careful management of parameterizations.
• MCMC methods such as sphere-walk Metropolis alleviate simplex boundary pathologies for Bayesian inference.
• ABC and forward simulation bypass likelihood computation, offering consistency at the expense of computational cost, especially when posterior mass is highly concentrated (Grazian et al., 2017).
• Policy design application of the Wallenius law is sensitive to regime constraints (e.g., assumptions on queue non-emptiness), and tightness of approximations may fail for stringent limits or on the feasible region boundary (Lin et al., 29 Jan 2025).

7. References and Context in the Statistical Literature

Principal references are Chesson (1976) for the integral-PMF representation, Fog (2008) for computational methods and asymptotic expansions, and Marin-Robert-Pudlo (2012) for ABC methodology (Grazian et al., 2017, Haines, 11 Nov 2025, Lin et al., 29 Jan 2025). Recent research has extended these foundations to complex contingency analysis, preference modeling, and stochastic scheduling, with new computational techniques (notably SWM) improving frequentist and Bayesian inference. The Wallenius distribution continues to serve as a critical tool for biased sampling, multi-category allocation, and robust inference under allocation bias.