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.

翻译：由于吉布斯点过程的归一化常数/配分函数普遍难以处理，推导其精确密度函数一直具有挑战性。本文利用点过程密度的一种近期替代表示，为这一开放问题提供了解决方案。对于有限点过程，该密度被表示为空间概率乘以高阶帕潘杰卢条件强度函数。通过利用依赖稀释的最新结果，推导了局部稳定点过程的生成泛函和空间概率的精确表达式。因此，也获得了密度/似然函数、配分函数和后验密度的精确表达式。最后，通过将格点上的局部稳定吉布斯随机场表示为点过程，将结果推广至该情形。