stylistic:tone, quality:correction
Subjective logic

Bayes' theorem represents a special case of deriving inverted conditional opinions in subjective logic expressed as:

where ϕ ~ {\displaystyle {\widetilde {\phi }}} denotes the operator for inverting conditional opinions. The argument (ω B | A S, ω B | ¬ A S) {\displaystyle (\omega _{B\vert A}^{S},\omega _{B\vert \lnot A}^{S})} denotes a pair of binomial conditional opinions given by source S {\displaystyle S}, and the argument a A {\displaystyle a_{A}} denotes the prior probability (aka. the base rate) of A {\displaystyle A}. The pair of derivative inverted conditional opinions is denoted (ω A | ~ B S, ω A | ~ ¬ B S) {\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})}. The conditional opinion ω A | B S {\displaystyle \omega _{A\vert B}^{S}} generalizes the probabilistic conditional P (A | B) {\displaystyle P(A\vert B)}, i.e. in addition to assigning a probability the source S {\displaystyle S} can assign any subjective opinion to the conditional statement (A | B) {\displaystyle (A\vert B)}. A binomial subjective opinion ω A S {\displaystyle \omega _{A}^{S}} is the belief in the truth of statement A {\displaystyle A} with degrees of epistemic uncertainty, as expressed by source S {\displaystyle S}. Every subjective opinion has a corresponding projected probability P (ω A S) {\displaystyle P(\omega _{A}^{S})}. 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.