A New Representation of Ewens-Pitman's Partition Structure and Its Characterization via Riordan Array Sums
Jan Greve
TL;DR
This work reframes Ewens-Pitman's partition structure, parameterized by $0\le \alpha<1$ and $\theta>-\alpha$, as a non-extreme harmonic function on the Kingman graph and constructs it via the interpolation polynomial approach. By translating Newton/umbral interpolation into Sheffer sequences and exploiting the isomorphism with Riordan arrays, the authors derive a new explicit representation of the harmonic function in terms of factorial monomial symmetric functions $m^*_\lambda$, clearly separating $\alpha$-dependent sizes from $\theta$-dependent length effects. They further show that certain marginals of the partition structure admit exponential Riordan array representations, enabling the fundamental theorem of Riordan arrays (FTRA) to yield compact closed forms for summary statistics and estimators, all within a symbolic generating-function framework. The resulting approach not only unifies existing estimators under a principled algebraic-combinatorial method but also facilitates symbolic computation and potential extensions to other branching-graph models used in Bayesian nonparametrics and related fields.
Abstract
Ewens-Pitman's partition structure arises as a system of sampling consistent probability distributions on set partitions induced by the Pitman-Yor process. It is widely used in statistical applications, particularly in species sampling models in Bayesian nonparametrics. Drawing references from the area of representation theory of the infinite symmetric group, we view Ewens-Pitman's partition structure as an example of a non-extreme harmonic function on a branching graph, specifically, the Kingman graph. Taking this perspective enables us to obtain combinatorial and algebraic constructions of this distribution using the interpolation polynomial approach proposed by Borodin and Olshanski (The Electronic Journal of Combinatorics, 7, 2000). We provide a new explicit representation of Ewens-Pitman's partition structure using modern umbral interpolation based on Sheffer polynomial sequences. In addition, we show that a certain type of marginals of this distribution can be computed using weighted row sums of a Riordan array. In this way, we show that some summary statistics and estimators derived from Ewens-Pitman's partition structure can be obtained using methods of generating functions. This approach simplifies otherwise cumbersome calculations of these quantities often involving various special combinatorial functions. In addition, it has the added benefit of being amenable to symbolic computation.