Let be a statistical null hypothesis (it contains statements about the probability distribution of the observable data x). In the classical statistics, a p-value can be employed to test this null hypothesis, see for instance this post. In the Bayesian paradigm, the posterior distribution is used, however, if is a sharp hypothesis (it is formed by a set with measure zero), then the posterior probability of given the observed data is zero. Let be a posterior probability, it is clear that the following sentence is false
“” “ is impossible to occur, given x”.
That is, zero probability does not mean impossibility of the null hypothesis. In order to measure possibility and impossibility of a hypothesis, one need to use other measures, e.g.: e-value and s-value. The former is built under a Bayesian paradigm and the latter under the classical one.
The e-value and s-value (notations: and , respectively) have the same behavior: they are possibility measures rather than probability ones. They provide a degree of contradiction between the observed data x and the null hypothesis H and have the following interpretations:
1. “” “ does not contradict ”,
2. “” “ fully contradicts ”,
3. “” “ contradicts more than ”.
It is possible to have and for the very same data and hypothesis. It just means that the observed data bring information that does not contradict a hypothesis formed by a set of measure zero. For the s-value, if the maximum likelihood estimative lies in the null set, then . For the e-value, if the mode of the posterior probability lies in the null set, then . It is straightforward to show that either or , the same for the e-value, where is the negation of .
In order to accept/reject a hypothesis H (assuming that the universe of hypotheses is closed), one should compute the s/e-value for the negation of H, that is
4. if and , one can accept if “” is sufficient small
5. if and , one can reject is “” is sufficient small
6. if a (or b) is not sufficient small, then more data are necessary to have a decision.
By this prescription, one will never accept a hypothesis formed by a set of Lebesgue measure zero (for both the s- and e-values).
Pereira, CAB, Stern, J., Wechsler, S. (2008). Can a significance test be genuinely Bayesian?, Bayesian Analysis, 3, 1, 79-10.
Patriota, AG. (2013). A classical measure of evidence for general null hypotheses, Fuzzy sets and Systems, 233, 74-88.
see also the comments in the blog