Subset Selection in Multinomial Distributions

• Subset selection for multinomial distributions is a method for choosing a representative set of outcomes that maximize informativeness and streamline statistical models.
• It employs convex relaxations, information-theoretic MDL, and large deviation theory to tackle NP-hard combinatorial challenges in high-dimensional settings.
• Practical applications span experimental design, feature selection in machine learning, and adaptive simulation allocation to ensure robust statistical inference.

Subset selection for multinomial distributions addresses the problem of optimally or efficiently choosing a subset of outcomes/categories (or the associated features, variables, or alternatives) from a multinomial model. The overarching goal is to maximize statistical informativeness, reduce model complexity, ensure robust inference, or practically constrain decision-making within a probabilistic, combinatorial, or Bayesian framework. Techniques span convex relaxations, information-theoretic criteria, combinatorial constructions, large deviation theory, categorical probabilistic models, advanced simulation-based allocation, and specialized approaches in high-dimensional or correlated data regimes.

1. Problem Formulation and Theoretical Foundations

Subset selection for multinomial distributions can be viewed as the task of identifying a subset of outcomes or features that together best represent, discriminate, or encapsulate the structure in the data where the underlying distribution of observed counts follows a multinomial law. In regression or classification (e.g., feature selection), one approach models outcome probabilities $p \in \Delta^m$ (the $m$-dimensional simplex) and wishes to select a subset $S \subset [m]$ of size $k$ maximizing a suitable objective under probabilistic constraints.

This problem is naturally combinatorial and often NP-hard due to $\ell_0$-type cardinality constraints (Bach et al., 2010). In experimental design, the objective may be to maximize the determinant of a Fisher information submatrix (D-optimal design) when the data model is multinomial (Wang et al., 2017). In ranking and selection or discrete multi-label learning, the goal may be to maximize the probability that the true best-performing alternative is selected or covered by the candidate subset (Painsky, 7 Aug 2025).

Formally, several canonical formulations arise, including:

• $\min_{\mathbf{w}} ||y - X\mathbf{w}||^2$ subject to $\operatorname{Card}(\mathbf{w}) \leq k$ (with counts modeled in $y$) (Bach et al., 2010)
• $\max_{S \subseteq [m], |S|=k} \det(I(\theta)_S)$, with $I(\theta)_S$ the Fisher information on $S$ (Wang et al., 2017)
• $$\mathbb{P}(s \in \mathcal{I}_\delta(X^n)) \geq 1-\delta$$ for selection confidence (Painsky, 7 Aug 2025)

Theoretical analyses involve tools from convex analysis, large deviation theory, combinatorial enumeration, and high-dimensional statistics.

2. Convex Relaxations and Sparse Eigenvalue Approaches

A widely applicable paradigm for subset selection under multinomial models is the use of convex relaxations, especially in feature selection and design problems (Bach et al., 2010). The core idea is to relax the $\ell_0$ cardinality constraint (which is nonconvex) by introducing a positive semidefinite matrix variable $Z$ and convex constraints. Specifically:

$$\max \mathrm{Tr}(A Z) \quad \text{subject to} \quad \|Z\|_1 \leq k, \; \mathrm{Tr}(Z) = 1, \; Z \succeq 0$$

where $A$ is typically derived from the data covariance or Fisher information matrix, and $\|Z\|_1$ enforces sparsity. In multinomial settings, $Z$ may represent a relaxed or fractional allocation over outcomes.

The sparse eigenvalue relaxation captures the extreme value of $x^\top A x$ over $x$ with at most $k$ nonzero entries. Approximation ratios such as

$$\frac{k}{p} \lambda_{\max}(A) \leq \lambda_{\max}^k(A) \leq \lambda_{\max}(A)$$

provide quantifiable guarantees on relaxation quality.

Integration with branch-and-bound strategies allows discarding (fathoming) infeasible branches rapidly, as the convex relaxation provides sharp lower bounds. Adapting to multinomial constraints typically requires further normalization (to ensure probabilities sum to one) and enforcing non-negativity.

Randomization techniques may leverage the covariance $Z$ to generate candidate subsets: for example, sampling $z \sim N(0, Z)$ and selecting the indices with the $k$ largest entries. In multinomial contexts, the sampling step must respect the simplex constraint.

3. Information-Theoretic and Model Selection Approaches

The Minimum Description Length (MDL) principle offers an information-theoretic criterion for subset selection in multinomial distributions (Boullé et al., 2016). The enumerative two-part crude MDL code is particularly suitable: parameters are restricted to the finite set of maximum likelihood estimators on $n$ samples, i.e., all count vectors summing to $n$.

The overall codelength decomposes into:

$$L(\theta, x^n) = \log \binom{n+m-1}{m-1} + \log \frac{n!}{n_1! \cdots n_m!}$$

Comparison with Normalized Maximum Likelihood (NML) codes reveals that while the enumerative code has roughly doubled parametric complexity asymptotically, its stochastic (data-given-parameter) part is lower. Empirical studies confirm its superior compression for "balanced" (non-degenerate) data.

This approach generalizes naturally from Bernoulli to multinomial models, though combinatorial complexity increases rapidly with the number of categories $m$. The MDL principle provides a natural framework for automatic subset selection where the "best" model, and associated subset, minimize the total codelength.

From a Bayesian perspective, this procedure is equivalent to placing a uniform prior on the finite set of empirical parameters and mixing uniformly over all sequences matching the empirical counts. This interpretation further supports the application of enumerative MDL codes for practical subset selection in multinomial settings.

4. Algorithmic and Statistical Methods in High Dimensions

Several papers provide algorithmic advances for subset selection in multinomial regimes:

• Polynomial-time computable triangular array representations for multinomial distributions organize the sample space into dyadic intervals almost uniquely determined by multinomial frequency profiles (Dobric et al., 2016). Admissible permutations enable bijections between such subsets and quantile representations, important in strong versions of the Central Limit Theorem (CLT).
• In discriminative settings, variable selection is integrated with probabilistic representation: vsLDA (Kim et al., 2012) introduces indicator variables to select informative words in topic modeling, restricting multinomial distributions for topics to informative subsets of the vocabulary. This achieves better performance in likelihood, stability, and classification compared to generic LDA, with a two-stage inference combining Gibbs sampling and Metropolis–Hastings for selection.
• In multi-label classification and subset prediction, cascade-based extensions of the Naive Bayes classifier (e.g., NaiBX) first predict the cardinality and then choose elements sequentially, reducing the joint estimation to a series of efficiently estimated univariate conditionals (Mossina et al., 2017). This matches or outperforms conventional multi-label methods in both time and accuracy, especially in large-alphabet multinomial settings.

5. Large Deviation Theory and Statistical Inference for Subsets

Large deviation results for multinomial distributions underpin thresholding rules and error quantification for subset selection—especially in high-dimensional and sparse regimes (Mirakhmedov, 2022). For statistics of the form $R_N(n) = \sum_{m=1}^N h_m(n_m)$, including power-divergence statistics (CR$_N(d)$), sharp asymptotic bounds for the probabilities of significant deviation are available:

$$P\left\{ R_N(n) > A_N + x\sigma_N \right\} = (1-\Phi(x)) \exp\{M(x)\}(1+o(1))$$

with expansions in skewness and variance tied to the number of cells $N$ and sparsity of $p_m$.

For subset selection, these results are central for:

• Setting thresholds to select cells with anomalous counts (signal detection)
• Determining confidence intervals for frequencies
• Managing sparse or "large-alphabet" multinomial problems, where standard normal or chi-square approximations may fail

In the specific setting of best-performer selection among competing algorithms, subset selection is formulated as covering the maximal win-probability symbol with high confidence (Painsky, 7 Aug 2025). Finite-sample and asymptotic results explicitly bound the minimal width $D_\delta(X^n)$ or $T_\delta(X^n)$ such that the set

$$\mathcal{I}_\delta(X^n) = \{ u : \hat{p}_u \geq \hat{p}_{[1]} - D_\delta(X^n) \}$$

contains the true best symbol with probability $\geq 1-\delta$. These results match established lower bounds, demonstrating near-optimality.

6. Categorical, Experimental Design, and Advanced Modeling Perspectives

Abstract categorical probability theory provides a generalization of subset selection through the Markov category formalism, defining multinomial and hypergeometric distributions as functors on the category of multisets (Jacobs, 2021). In this context:

• Multinomial draws are modeled as copying followed by projection and aggregation.
• Hypergeometric draws (subset selection without replacement) arise via repeated delete operations, formalizing classical probabilistic subset selection (urn) procedures in a categorical setting.

Experimental design approaches, such as D-optimal selection via determinantal point processes (DPP), emphasize maximizing the determinant of Fisher information or covariance matrices over all possible $k$-subsets (Wang et al., 2017). The k-DPP samples subsets with probability proportional to their determinant, offering a polynomial-time, randomized (approximation) algorithm to explore the combinatorial space efficiently.

Simulation-driven subset selection (as in ranking and selection with regression metamodels) leverages metamodels (often quadratic approximations) to allocate simulation budgets efficiently across alternatives, given multinomial estimation noise (Gao et al., 2019). Optimal allocation is derived via large deviations analysis, partitioning the solution space as needed to fulfill local approximations.

The Conway-Maxwell-multinomial (CMM) distribution (Morris et al., 2019) generalizes the multinomial to accommodate both positive and negative association among counts, introducing a dispersion parameter. Importantly, conditionals of CMM on subsets remain in (conjugate) tractable families (e.g., the Conway-Maxwell-binomial), supporting robust inference for category groupings and flexible modeling in real data with over- or under-dispersion.

Finally, in subset selection under alternative sampling in multinomial logit models, McFadden’s correction factor is necessary and sufficient to ensure that consistent (unbiased) estimates are obtained when only a subset of alternatives is observed, minimizing the expected information loss with respect to the parameters of interest (Dekker et al., 2021).

7. Practical Applications and Implications

Applications of subset selection for multinomial distributions span topic modeling (automatic vocabulary subset selection), experimental design (choosing informative treatments or sensor locations), optimal algorithm selection, adaptive simulation budget allocation, and multi-label learning.

Practical procedures are characterized by tradeoffs in statistical fidelity, computational efficiency, and robustness:

• Convex relaxations and eigenvalue-based SDPs enable scalable computation and tight bounding for large-scale feature selection and sparse modeling.
• Information-theoretic and MDL-based criteria offer parameterization-invariant, automatically complexity-regularized subset selection, with enumeration-based codes providing practical advantages in balanced data.
• Randomized and DPP-inspired subset samplers efficiently scale to massive combinatorial spaces and approach optimality given sufficient iterations.
• Large-deviation and finite-sample theory underpin error control and confidence thresholding, essential in settings with many rare outcomes or close alternatives.
• Bayesian and categorical frameworks establish principled, generalizable subset selection suitable for structured and uncertain probabilistic environments.

Subset selection for multinomial distributions thus forms a foundational class of methods with deep theoretical guarantees and a breadth of operational methods for discrete, high-dimensional, and complex data.