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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 ) 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 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.