e-value and s-value: possibility rather than probability measures

Let H 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 H is a sharp hypothesis (it is formed by a set with measure zero), then the posterior probability of H given the observed data is zero. Let \pi(.|x) be a posterior probability, it is clear that the following sentence is false

\pi(H|x)=0\Rightarrow  “H 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: ev(.|x) and s(.|x), 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. “s(H|x) = 1\Rightarrowx does not contradict H”,
2. “s(H|x) = 0\Rightarrowx fully contradicts H”,
3. “s(H'|x) < s(H''|x)\Rightarrowx contradicts more H' than H''”.

It is possible to have s(H|x) = ev(H|x) = 1 and \pi(H|x) = 0 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 s(H|x) = 1. For the e-value, if the mode of the posterior probability lies in the null set, then ev(H|x) =1. It is straightforward to show that either s(H|x) = 1 or s(\neg H|x) = 1, the same for the e-value, where \neg H is the negation of H.

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 s(H|x) = 1 and s(\neg H| x) = a, one can accept H if “a” is sufficient small
5. if s(H|x) = b and s(\neg H|x) = 1, one can reject H is “b” 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


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