beta-binomial approximation

This study investigates the capacity of a binary channel with continuous input and discrete output, along with the structure of its optimal input distribution. Through information-theoretic analysis, convex optimization, and Beta-Binomial modeling—augmented by minimax redundancy constructions and divergence measures including relative entropy and χ² divergence—the authors prove that the optimal input distribution is discrete, unique, symmetric about 1/2, and includes the endpoints of the interval. Key contributions include tightening the upper bound on the support size of the optimal input from O(n) to O(n/2) and establishing the first lower bound of Ω(√(n log log n)). They further show that the output distribution induced by Beta(1/2,1/2) is asymptotically optimal, derive non-asymptotic capacity bounds, and establish that C(n) = ½ log(nπ/(2e)) + o(1), with numerical experiments corroborating the theoretical findings.

This study investigates a tight lower bound on the support size of input distributions that achieve the capacity of the binomial channel. By analyzing the structure of the output distribution and leveraging the asymptotic optimality of the Beta-binomial distribution, the authors establish a refined approximation linking channel capacity to the Beta-binomial law, employing tools from information-theoretic capacity analysis, relative entropy, and χ²-divergence comparisons. The main contribution is an improvement of the known lower bound on the support size from √n to the order of √(n log log n), proving that any capacity-achieving input distribution must contain at least this many mass points. Additionally, the paper provides an asymptotic expression for the channel capacity: C(n) = ½ log(nπ/2e) + o(1).

This work addresses the lack of effective randomness tests for broad classes of probability distributions by proposing a novel framework termed *e-variable-approximability*. For the first time, this approach enables the construction of computable e-variables for common distribution classes. By integrating Levin’s notion of randomness tests with approximation techniques, the method yields explicit and computable randomness tests tailored to entire distribution families. It provides both theoretical guarantees and practical testing procedures for important classes such as exponential families, thereby substantially extending the applicability of e-variables in algorithmic information theory and statistical inference.

This paper addresses high-probability estimation of discrete distributions over a finite alphabet under the Kullback–Leibler (KL) divergence, focusing on the sparse regime where the sample size is smaller than the alphabet size. We propose a data-driven, adaptive smoothing estimator tailored to this setting. Our main contributions are threefold: First, we establish the first non-asymptotic, high-probability upper bound on the KL risk, which is minimax optimal up to constant factors. Second, we derive a sharp high-probability upper bound on the missing mass. Third, we prove that the KL risk of the Laplace estimator admits tight high-probability upper and lower bounds; that the minimax high-probability risk incurs an extra logarithmic factor; and—under a sparsity assumption—that the risk bound depends only on two effective sparsity parameters, enabling automatic adaptation to underlying distribution structure. Collectively, these results unify and substantially extend the theoretical foundations of classical smoothing methods, delivering rigorous, practical risk guarantees for small-sample discrete distribution estimation.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This work addresses the challenge of efficiently and exactly sampling from Beta, Gamma, and Dirichlet distributions when their shape parameters are less than one—a regime where existing methods often require iterative or approximate procedures. The authors propose a novel approach based on explicit deterministic transformations that generates exact samples using only a fixed number of independent uniform random variables and elementary arithmetic operations. This method yields concise, non-iterative, and approximation-free “extended one-liners” for Beta(a,b) with min(a,b)<1, Gamma(c) with c<1, and Dirichlet(α₁,…,α_d) with 0<α_i<1. By eliminating the need for rejection sampling or numerical inversion, the scheme preserves mathematical exactness while significantly improving computational efficiency.

This study addresses the inadequacy of the standard binomial distribution in effectively modeling extreme values commonly observed in real-world discrete data. To overcome this limitation, the authors propose a novel heavy-tailed binomial-like distribution constructed via a weighted arithmetic average of the conventional binomial distribution and a newly devised double-uniform distribution. The resulting model retains structural simplicity while substantially enhancing tail probability representation. The paper comprehensively derives the distribution’s statistical properties, develops corresponding methods for parameter estimation and hypothesis testing, and demonstrates its superior fit and practical applicability on benchmark datasets featuring heavy-tailed discrete observations. This work exemplifies a complete statistical modeling paradigm, spanning theoretical formulation to empirical validation.

This work addresses the challenge of modeling multivariate count data with exclusion or incompatibility constraints on graph-structured variables by proposing a unified graphical distribution framework. It systematically constructs, for the first time, four families of distributions—graphical multinomial, negative multinomial, hypergeometric, and negative hypergeometric—leveraging decomposable graphs to encode variable dependencies and feasible configurations. The framework supports continuous interpolation between the empty and complete graphs while preserving an explicit Markov factorization and tractable sampling schemes. Building upon graphical Dirichlet-type priors, a Bayesian hierarchical model is developed, yielding closed-form posterior and predictive distributions. The approach demonstrates both flexibility and practical utility in constrained counting tasks, such as Rydberg atom excitation experiments.

This study addresses the Kiefer–Weiss problem in a nonparametric setting, aiming to minimize a weighted sum of error probabilities in binary sequential hypothesis testing under a constraint on the worst-case expected sample size. By reformulating the problem as an optimal stopping problem, the authors introduce finitely randomized strategies and take their limit to derive, for the first time, a complete solution. The resulting optimal stopping rule is characterized by a two-dimensional statistic comprising the likelihood ratio and the remaining allowable sample size. A key innovation lies in uncovering a novel mechanism—dynamic sample-size adjustment via randomization—that enhances detection performance. Practical approximate strategies are also proposed. The methodology is applicable to both Bernoulli success probability testing and normal mean shift detection, with numerical experiments confirming its efficacy.