  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.
