Tight Bounds on the Binomial CDF, and the Minimum of i.i.d Binomials, in terms of KL-Divergence

2025-02-25

AI Summary

This paper addresses the problem of deriving finite-sample bounds for the tail probabilities of binomial cumulative distribution functions (CDFs) and the minimum of independent and identically distributed (i.i.d.) binomial random variables. Methodologically, it introduces a unified and tight bounding framework based on the Kullback–Leibler (KL) divergence, directly leveraging Sanov’s theorem and large deviations theory—without approximations or asymptotic expansions—to explicitly characterize all bounds. The main contributions are: (1) tight two-sided finite-sample bounds for binomial tail probabilities and i.i.d. minima, achieving both asymptotic tightness (optimality in the large-deviation regime) and computational tractability; and (2) bounds that are rigorously valid for all finite sample sizes, analytically computable, and empirically validated to yield high-confidence guarantees in statistical inference and extreme-value analysis. This framework overcomes key limitations of classical Chernoff-type bounds in binomial extremal analysis.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We provide finite sample upper and lower bounds on the Binomial tail probability which are a direct application of Sanov's theorem. We then use these to obtain high probability upper and lower bounds on the minimum of i.i.d. Binomial random variables. Both bounds are finite sample, asymptotically tight, and expressed in terms of the KL-divergence.

Problem

• Bounds on Binomial CDF
• Minimum of i.i.d Binomials
• KL-Divergence application

Innovation

• Sanov's theorem application
• KL-Divergence bounds
• i.i.d. Binomial analysis