# Distributed Hypothesis Testing Over Noisy Channels

## TL;DR

This paper investigates distributed binary hypothesis testing over noisy channels, deriving bounds on the error exponent and revealing that optimal performance depends on marginal distributions, with no separation between hypothesis testing and channel coding.

## Contribution

It provides single-letter bounds on the type 2 error exponent, characterizes the optimal exponent for single helpers, and shows the importance of data-channel correlation.

## Key findings

- Bounds on the type 2 error exponent are derived.
- Optimal exponent depends on marginal distributions, not joint.
- Operational separation between hypothesis testing and channel coding does not hold.

## Abstract

A distributed binary hypothesis testing problem, in which multiple observers transmit their observations to a detector over noisy channels, is studied. Given its own side information, the goal of the detector is to decide between two hypotheses for the joint distribution of the data. Single-letter upper and lower bounds on the optimal type 2 error exponent (T2-EE), when the type 1 error probability vanishes with the block-length are obtained. These bounds coincide and characterize the optimal T2-EE when only a single helper is involved. Our result shows that the optimal T2-EE depends on the marginal distributions of the data and the channels rather than their joint distribution. However, an operational separation between HT and channel coding does not hold, and the optimal T2-EE is achieved by generating channel inputs correlated with observed data.