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Bernstein-Type Bounds for Beta Distribution

Maciej Skorski

Abstract

This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-gaussian and sub-gamma bounds previously studied in this context. The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

Bernstein-Type Bounds for Beta Distribution

Abstract

This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-gaussian and sub-gamma bounds previously studied in this context. The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

Paper Structure

This paper contains 21 sections, 9 theorems, 50 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Contribution
  4. Related Work
  5. Organization
  6. Results
  7. Optimal Bernstein-Type Concentration Bounds
  8. Handy Recurrence for Central Moments
  9. Numerical Evaluation
  10. Preliminaries
  11. Gaussian Hypergeometric Function
  12. Beta Distribution Properties
  13. Cramér-Chernoff method
  14. Logarithmic Inequalities
  15. Proofs
  16. ...and 6 more sections

Key Result

theorem 1

Let $X\sim\mathsf{Beta}(\alpha,\beta)$. Define the parameters Then the upper tail of $X$ is bounded as and the lower tail of $X$ is bounded as

Figures (3)

  • Figure 1: Numerical evaluation of the best sub-gamma bounds (\ref{['thm:beta_bernstein']}) and the best sub-gaussian bounds (marchal2017sub).
  • Figure 2: The logarithmic inequality from \ref{['lemma:log_ineq']}.
  • Figure 3: The proof roadmap.

Theorems & Definitions (19)

  • remark 1
  • theorem 1: Bernstein's Inequality
  • remark 2: Variance and Scale Parameters
  • remark 3: Mixed Gaussian-Exponential Behaviour
  • theorem 2: Best Cramér-Chernoff Bound
  • corollary 1: Optimality of \ref{['thm:beta_bernstein']}
  • theorem 3: Order 2 Recurrence for Central Moments
  • corollary 2: P-recursive Property
  • corollary 3: Skewness of Beta Distribution
  • lemma 1: Hypergeometric Contiguous Recurrence
  • ...and 9 more