Evidence for Alternative Hypotheses

Most researchers want evidence for the direction of an effect, not evidence against a point null hypothesis. Such evidence is ideally on a scale that is easily in- terpretable, with an accompanying standard error. Further, the evidence from iden- tical experiments should be repeatable, and evidence from independent experiments should be easily combined, such as required in meta-analysis. Such a measure of evidence exists and has been shown to be closely related to the Kullback-Leibler symmetrized distance between null and alternative hypotheses for exponential fam- ilies. Here we provide more examples of the latter phenomenon, for distributions ly- ing outside the class of exponential families, including the non-central chi-squared family with unknown non-centrality parameter.