a problem that did not need ABC in the end

While in Denver, at JSM, I came across [across validated!] this primarily challenging problem of finding the posterior of the 10³ long probability vector of a Multinomial M(10⁶,p) when only observing the range of a realisation of M(10⁶,p). This sounded challenging because the distribution of the pair (min,max) is not available in closed form. (Although this allowed me to find a paper on the topic by the late Shanti Gupta, who was chair at Purdue University when I visited 32 years ago…) This seemed to call for ABC (especially since I was about to give an introductory lecture on the topic!, law of the hammer…), but the simulation of datasets compatible with the extreme values of both minimum and maximum, m=80 and M=12000, proved difficult when using a uniform Dirichlet prior on the probability vector, since these extremes called for both small and large values of the probabilities. However, I later realised that the problem could be brought down to a Multinomial with only three categories and the observation (m,M,n-m-M), leading to an obvious Dirichlet posterior and a predictive for the remaining 10³-2 realisations.