Jan Greve - The use of Riordan arrays for the prior likelihood conflict in species sampling models

We introduce a tool developed in combinatorics called the Riordan array in a problem called “prior-likelihood conflict” in Bayesian clustering.
Various Bayesian clustering models that tackle the species sampling problems are equipped with a prior distribution on partitions of n data which is a finite-dimensional distribution of a stochastic process on the Young lattice. This distribution can be considered as the set of all Markov processes of central growth of Young diagrams starting at the empty set and terminated after the n-th step. The probability of such a process depends only on the shape of the terminating Young diagram.

Various limits of this process have been studied as an asymptotic combinatorial problem and it is known that for sufficiently large n, all such processes terminate at Young diagrams of similar shape. For this reason, the prior distribution on partitions of n is necessarily biased, sometimes to such an extent that information from the likelihood is hardly incorporated into the posterior. To account for this conflict, we introduce Riordan arrays to facilitate model-free and efficient computation of various summary statistics on the aforementioned prior distribution to assess the extent of biasedness in the prior.