Elementary rules of Bayesian probability theory

Bayesian probability theory can be conveniently summarised in the following elementary rules:

Sum rule:

$P(A \vert B) + P(\neg A \vert B) = 1$

Product rule:

P(AB | C) = P(A | C) P(B | AC)

Here P(A | B) denotes the probability of A on the condition that B is true. These rules correspond to the negation and conjunction operations of Boolean algebra. The disjunction does not need a separate rule because it can be derived from negation and conjunction: $A + B = \neg(\neg A \neg B)$. In fact, only one operation would suffice since other operations can be derived from either NAND or NOR operation alone. The NAND operation, for instance, yields the following rule, starting from which every other rule of Bayesian probability theory can be derived: $P(\neg A + \neg B \vert C) + P(A \vert C) P(B \vert AC) = 1$.

These rules fix the scale on which the degrees of belief are measured. Cox showed that under very general requirements of consistency and compatibility with common sense, the rules of calculus with beliefs have to be homomorphic with the sum and product rule [19]. This means that one can measure the degrees of beliefs on any scale, but it is possible to transform the degrees of beliefs on the canonical scale of probabilities such that the rules for negation and conjunction take the form of the sum and product rule.