can be found in Jaynes' book, but it does admit an easier to understand alternative derivation, once the solution is known. Another point to emphasise is that the prior state of knowledge described by the rule of succession is given as an enumeration of the possibilities, with the additional information that it is possible to observe each category. This can be equivalently stated as observing each category once prior to gathering the data. To denote that this is the knowledge used, an Im is put as part of the conditions in the probability assignments. The rule of succession comes from setting a binomial likelihood, and a uniform prior distribution. Thus a straightforward generalisation is just the multivariate extensions of these two distributions: 1) Setting a uniform prior over the initial m categories, and 2) using the multinomial distribution as the likelihood function (which is the multivariate generalisation of the binomial distribution). It can be shown that the uniform distribution is a special case of the Dirichlet distribution with all of its parameters equal to 1 (just as the uniform is Beta(1,1) in the binary case). The Dirichlet distribution is the conjugate prior for the multinomial distribution, which means that the posterior distribution is also a Dirichlet distribution with different parameters. Let pi denote the probability that category i will be observed, and let ni denote the number of times category i (i = 1, ..., m) actually was observed. Then the joint posterior distribution of the probabilities p1, ..., pm is given by: To get the generalised rule of succession, note that the probability of observing category i on the next observation, conditional on the pi is just pi, we simply require its expectation. Letting Ai denote the event that the next observation is in category i (i = 1, ..., m), and let n = n1 + ... + nm be the total number of observations made. The result, using the properties of the Dirichlet distribution is: This solution reduces to the probability that would be assigned using the principle of indifference before any observations made (i.e. n = 0), consistent with the original rule of succession. It also contains the rule of succession as a special case, when m = 2, as a generalisation should. Because the propositions or events Ai are mutually exclusive, it is possible to collapse the m categories into 2. Simply add up the