Applying Bayes' Theorem in Deduction
Correspondence to other mathematical frameworks
Propositional logic
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:
Subjective logic
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.