Model fusion and multiple testing in the likelihood paradigm: shrinkage and evidence supporting a point null hypothesis

According to the general law of likelihood, the strength of statistical evidence for a hypothesis as opposed to its alternative is the ratio of their likelihoods, each maximized over the parameter of interest. Consider the problem of assessing the weight of evidence for each of several hypotheses. Under a realistic model with a free parameter for each alternative hypothesis, this leads to weighing evidence without any shrinkage toward a presumption of the truth of each null hypothesis. That lack of shrinkage can lead to many false positives in settings with large numbers of hypotheses. A related problem is that point hypotheses cannot have more support than their alternatives. Both problems may be solved by fusing the realistic model with a model of a more restricted parameter space for use with the general law of likelihood. Applying the proposed framework of model fusion to data sets from genomics and education yields intuitively reasonable weights of evidence.