When conditional probabilities \(P\left(H|E\right)\) are defined through the ratio formula, conditionalization satisfies two intuitively appealing iteration principles: Commutativity (the order of iterated conditionalization does not affect the final outcome) and Accumulation (iterated conditionalization yields the same result as conditionalizing upon the conjunction of the individual conditioning propositions). When P(E) = 0, the ratio formula is ill-defined, so a more general treatment of conditional probability is needed. The standard mathematical treatment, stemming from Kolmogorov’s work, centers upon regular conditional distributions (rcds). I investigate the degree to which Commutativity and Accumulation extend to the general setting provided by rcds. In the right circumstances, rcd conditionalization turns out to satisfy iteration principles analogous to Commutativity and Accumulation. However, the iteration principles usually allow exceptions. Moreover, there are circumstances where rcd conditionalization satisfies nothing resembling Commutativity or Accumulation. Grappling with these complexities is essential for assessing how well rcds serve Bayesian inference.
