New density/likelihood representations for Gibbs models based on generating functionals of point processes

TL;DR

This paper introduces new density and likelihood representations for Gibbs point processes using generating functionals, enabling exact calculations of densities, partition functions, and posterior densities, which were previously intractable.

Contribution

It provides a novel approach to express Gibbs process densities via void probabilities and Papangelou intensities, solving a longstanding intractability problem.

Findings

Derived exact expressions for densities and likelihoods of Gibbs point processes.

Extended the approach to Gibbsian random fields on lattices.

Connected point process representations to Gibbsian models on lattices.

Abstract

Deriving exact density functions for Gibbs point processes has been challenging due to their general intractability, stemming from the intractability of their normalising constants/partition functions. This paper offers a solution to this open problem by exploiting a recent alternative representation of point process densities. Here, for a finite point process, the density is expressed as the void probability multiplied by a higher-order Papangelou conditional intensity function. By leveraging recent results on dependent thinnings, exact expressions for generating functionals and void probabilities of locally stable point processes are derived. Consequently, exact expressions for density/likelihood functions, partition functions and posterior densities are also obtained. The paper finally extends the results to locally stable Gibbsian random fields on lattices by representing them as point processes.