Polylogarithm Models for Heavy-Tailed Counts: Approximate Bayesian Inference and Application to Ecological Networks

Polylogarithm Models for Heavy-Tailed Counts: Approximate Bayesian Inference and Application to Ecological Networks

Luiza Piancastelli, University College Dublin.

16th of October 2025

Abstract: 

Ecological surveys often generate count data to assess biodiversity and ecosystem dynamics, capturing species abundances or behavioural interactions. In this study, we analyse flower–invertebrate interactions in the Brazilian Atlantic that span four plant orders and 28 invertebrate orders. Our main objectives are to model the distribution of interaction frequencies within the resulting network and to cluster flower–invertebrate interactions. These tasks are complicated by the heavy-tailed nature of the data, which is a key modelling challenge we address. To this end, we provide a complete toolkit for sampling and fitting the Polylogarithm distribution, a discrete distribution capable of capturing heavy-tailed behaviour. We show that the Polylogarithm distribution, in some sense, unifies several well-known discrete heavy-tailed distributions, but this comes at the cost of an intractable probability mass function. The challenging task of generating exact draws from the distribution is addressed through the design of a rejection sampler, which overcomes the unavailable terms. This sampler then serves as the basis for constructing a Bayesian inferential framework via Approximate Bayesian Computation (ABC). Finally, a finite mixture generalisation of the Polylogarithm distribution is explored as a means to reveal ecological interaction patterns.