Should unfolded histograms be used to test hypotheses?

In many analyses in high energy physics, attempts are made to remove the effects of detector smearing in data by techniques referred to as "unfolding" histograms, thus obtaining estimates of the true values of histogram bin contents. Such unfolded histograms are then compared to theoretical predictions, either to judge the goodness of fit of a theory, or to compare the abilities of two or more theories to describe the data. When doing this, even informally, one is testing hypotheses. However, a more fundamentally sound way to test hypotheses is to smear the theoretical predictions by simulating detector response and then comparing to the data without unfolding; this is also frequently done in high energy physics, particularly in searches for new physics. One can thus ask: to what extent does hypothesis testing after unfolding data materially reproduce the results obtained from testing by smearing theoretical predictions? We argue that this "bottom-line-test" of unfolding methods should be studied more commonly, in addition to common practices of examining variance and bias of estimates of the true contents of histogram bins. We illustrate bottom-line-tests in a simple toy problem with two hypotheses.