Homeopathic priors?

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.