Efficient computation of tight approximations to Chernoff bounds

TL;DR

This paper introduces new quadratic-invertible approximations to Chernoff bounds that are more accurate than previous methods, facilitating easier and tighter tail probability bounds in probability distributions.

Contribution

The paper presents novel approximations to Chernoff bounds that can be inverted exactly via quadratic equations, improving accuracy over existing approximations.

Findings

New quadratic-invertible Chernoff bound approximations

Improved accuracy over previous approximation methods

Facilitates tighter tail probability bounds

Abstract

Chernoff bounds are a powerful application of the Markov inequality to produce strong bounds on the tails of probability distributions. They are often used to bound the tail probabilities of sums of Poisson trials, or in regression to produce conservative confidence intervals for the parameters of such trials. The bounds provide expressions for the tail probabilities that can be inverted for a given probability/confidence to provide tail intervals. The inversions involve the solution of transcendental equations and it is often convenient to substitute approximations that can be exactly solved e.g. by the quadratic equation. In this paper we introduce approximations for the Chernoff bounds whose inversion can be exactly solved with a quadratic equation, but which are closer approximations than those adopted previously.