One of the common ways to use conditional probability is through Bayes' Theorem. The definition of conditional probability is used in Bayes' Theorem to render inferences in many situations where events are causally linked.
Using twice, one may use Bayes' theorem to also express in terms of and without negations:
when . From this we can read off the inference
In words: If certainly implies , we infer that certainly implies . Where , the two implications being certain are equivalent statements. In the probability formulas, the conditional probability generalizes the logical implication , where now beyond assigning true or false, we assign probability values to statements. The assertion of is captured by certainty of the conditional, the assertion of . Relating the directions of implication, Bayes' theorem represents a generalization of the contraposition law, which in classical propositional logic can be expressed as:
In this relation between implications, the positions of resp. get flipped.
The corresponding formula in terms of probability calculus is Bayes' theorem, which in its expanded form involving the prior probability/base rate of only , is expressed as:
Bayes' theorem represents a special case of deriving inverted conditional opinions in subjective logic expressed as:
where denotes the operator for inverting conditional opinions. The argument denotes a pair of binomial conditional opinions given by source , and the argument \(a_{A}} denotes the prior probability (aka. the base rate) of . The pair of derivative inverted conditional opinions is denoted . The conditional opinion generalizes the probabilistic conditional , i.e. in addition to assigning a probability the source can assign any subjective opinion to the conditional statement . A binomial subjective opinion is the belief in the truth of statement with degrees of epistemic uncertainty, as expressed by source . Every subjective opinion has a corresponding projected probability . The application of Bayes' theorem to projected probabilities of opinions is a homomorphism, meaning that Bayes' theorem can be expressed in terms of projected probabilities of opinions:
Hence, the subjective Bayes' theorem represents a generalization of Bayes' theorem.