**8.5 Application
Of The Probability Theory**

In logical expression, when e implies h, that is, e Æ h, can be alternatively read as 'e is sufficient for h' or as 'h is necessary for e'. There is no ambiguity between e and h, i.e., the reliability is 100%. However, in reality, the reliability of e in support of h is lower than a logical implication.

8.5.1. Necessity and Sufficiency Measures

8.5.2. Posterior Probability Estimation

8.5.3. Maximum Likelihood Decision Rule Based on Penalty Functions