# From Blackwell Dominance in Large Samples to Renyi Divergences and Back Again

## TL;DR

This paper establishes that in large samples, the informativeness of experiments is characterized by Renyi divergences, linking Blackwell dominance to these divergences and showing that additive divergences satisfying data processing are integrals of Renyi divergences.

## Contribution

It provides a new characterization of Blackwell dominance in large samples using Renyi divergences and shows that additive divergences satisfying data processing are integrals of Renyi divergences.

## Key findings

- Blackwell dominance in large samples is characterized by Renyi divergences.
- Renyi divergences are additive with respect to product distributions.
- Every additive divergence satisfying data processing is an integral of Renyi divergences.

## Abstract

We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951), we show that generically an experiment is more informative than another in large samples if and only if it has higher Renyi divergences. We apply our analysis to the problem of measuring the degree of dissimilarity between distributions by means of divergences. A useful property of Renyi divergences is their additivity with respect to product distributions. Our characterization of Blackwell dominance in large samples implies that every additive divergence that satisfies the data processing inequality is an integral of Renyi divergences.