Homeopathic priors?

TL;DR

This paper investigates how different Bayesian priors affect the detection of weak signals in cluttered backgrounds, revealing that priors with infinitesimal weight can significantly influence detection performance.

Contribution

It introduces a family of priors with singular weights at zero target strength, highlighting their nontrivial effects on Bayesian detection in low-abundance scenarios.

Findings

Priors with singular weight at zero can improve detection of weak signals.

Infinitesimal weight components in priors have significant effects.

The approach enhances understanding of prior influence in composite hypothesis testing.

Abstract

The problem of composite hypothesis testing is considered in the context of Bayesian detection of weak target signals in cluttered backgrounds. (A specific example is the detection of sub-pixel targets in multispectral imagery.) In this model, the target strength (call it $a$) is an unknown parameter, and that lack of knowledge can be addressed by incorporating a prior over possible parameter values. The performance of the detector depends on the choice of prior, and -- with the motivation of enabling better performance at low target abundances -- a family of priors are investigated in which singular weight is associated with the $a\to 0$ limit. Careful treatment of this limiting process leads to a situation in which components of the prior with infinitesimal weight have nontrivial effects. Similar claims have been made for homeopathic medicines.