Large deviations, asymptotic bounds on the number of positive individuals in a Bernoulli sample via the number of positive pool samples drawn on the bernoulli sample

TL;DR

This paper establishes large deviation principles for empirical infection measures in Bernoulli samples and pools, revealing asymptotic relationships between individual infection proportions and pooled sample infection proportions.

Contribution

It introduces joint large deviation principles for empirical infection measures and derives asymptotic relationships between individual and pooled infection proportions.

Findings

Established joint large deviation principles for infection measures.

Derived asymptotic relationships between individual and pooled infection proportions.

Expressed rate functions in terms of relative entropies.

Abstract

In this paper we define for a Bernoulli samples the \emph{ empirical infection measure}, which counts the number of positives (infections) in the Bernoulli sample and for the \emph{ pool samples} we define the empirical pool infection measure, which counts the number of positive (infected) pool samples. For this empirical measures we prove a joint large deviation principle for Bernoulli samples. We also found an asymptotic relationship between the \emph{ proportion of infected individuals } with respect to the samples size, $n$ and the \emph{ proportion of infected pool samples} with respect to the number of pool samples, $k(n).$ All rate functions are expressed in terms of relative entropies.