Applying Bayes' Theorem in Deduction

Propositional logic

Using $P(\neg B\vert A)=1-P(B\vert A)$ twice, one may use Bayes' theorem to also express $P(\neg B\vert \neg A)$ in terms of $P(A\vert B)$ and without negations:

$P(\neg B\vert \neg A)=1-\left(1-P(A\vert B)\right){\frac {P(B)}{P(\neg A)}}$,

when $P(\neg A)=1-P(A)\neq 0$. From this we can read off the inference

$P(A\vert B)=1\implies P(\neg B\vert \neg A)=1$.

In words: If certainly $B$ implies $A$, we infer that certainly $\neg A$ implies $\neg B$. Where $P(B)\neq 0$, the two implications being certain are equivalent statements. In the probability formulas, the conditional probability $P(A\vert B)$ generalizes the logical implication $B\implies A$, where now beyond assigning true or false, we assign probability values to statements. The assertion of $B\implies A$ is captured by certainty of the conditional, the assertion of $P(A\vert B)=1$. Relating the directions of implication, Bayes' theorem represents a generalization of the contraposition law, which in classical propositional logic can be expressed as:

$(B\implies A)\iff (\neg A\implies \neg B)$.

In this relation between implications, the positions of $A$ resp. $B$ get flipped.

The corresponding formula in terms of probability calculus is Bayes' theorem, which in its expanded form involving the prior probability/base rate $a$ of only $A$, is expressed as:

$P(A\vert B)=P(B\vert A){\frac {a(A)}{P(B\vert A)\,a(A)+P(B\vert \neg A)\,a(\neg A)}}$.

Subjective logic

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

$(\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})=(\omega _{B\vert A}^{S},\omega _{B\vert \lnot A}^{S}){\widetilde {\phi }}a_{A}$,

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

$P(\omega _{A{\tilde {|}}B}^{S})={\frac {P(\omega _{B\vert A}^{S})a(A)}{P(\omega _{B\vert A}^{S})a(A)+P(\omega _{B\vert \lnot A}^{S})a(\lnot A)}}$.

Hence, the subjective Bayes' theorem represents a generalization of Bayes' theorem.