We develop an extension of continuum fusion methods that allows the generation of unbeatable decision rules for
discrete binary composite hypothesis testing problems. Background: Amongst the many flavors of continuum fusion (CF) algorithm, one can always be found that will produce the uniformly
most powerful (UMP) solution to any composite hypothesis (CH) testing problem, when such a solution exists . This
optimality property, combined with the flexibility in design afforded by CF principles, led to the prospect that with any
reasonably defined optimality metric, any detection problem could be solved with some CF-based decision rule (DR).
Doubt was cast on this possibility in a paper by Theiler , who showed that applying continuum fusion logical rules to
a particular discrete (as opposed to continuum) problem could not produce the better algorithm. Theiler’s example
requires creation of a CH test, and for these no generally optimal form exists. However, Theiler’s problem also obeyed
an invariance principle, and if solutions are restricted to obey the same invariance, then a uniformly most powerful
invariant (UMPI) solution does exist. This solution cannot be generated by applying current CF principles to this discrete
parameter problem. In short, standard CF logic cannot produce a highly desirable answer.
The UMPI solution exemplifies Bayesian solutions to discrete parameter CH problems, and it is shown below why
standard CF solutions cannot always produce them, in agreement with Theiler’s result. Bayesian solutions feature
prominently in statistical decision theory, because they form the class of unbeatable decision rules, as defined below.
Thus, standard CF principles cannot produce an important class of solutions to discrete CH problems.
Here we extend the CF methodology in a way that converts any discrete parameter fusion problem into a continuous one.
Continuum fusion solutions to the converted problem then generate the entire class of unbeatable detectors.