Probability Theory without Bayes' Rule
Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for probability theory, according to which the reverse conditional probability distribution P(B|A) is not specified by the forward conditional probability distribution P(A|B) and the marginals P(A) and P(B). Thus, in order to perform statistical inference, one must specify an additional "inference axiom," which relates P(B|A) to P(A|B), P(A), and P(B). We show that when Bayes' rule is chosen as the inference axiom, the axioms are equivalent to the classical Kolmogorov axioms. We then derive consistency conditions on the inference axiom, and thereby characterize the set of all possible rules for inference. The set of "first-order" inference axioms, defined as the set of axioms in which P(B|A) depends on the first power of P(A|B), is found to be a 1-simplex, with Bayes' rule at one of the extreme points. The other extreme point, the "inversion rule," is studied in depth.