  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.
