Bartlett identities link score, curvature, and Fisher information in classical likelihood theory, but no general formulation exists for spatial point processes, where likelihoods act on configuration space and parameters are functions. In this paper, we establish Bartlett identities for point processes by treating the log-likelihood as a functional and using variational derivatives. This yields a hierarchy of identities in which derivatives of the log-likelihood are represented by functions and kernels, and are expressed in terms of cumulants of the score field. As a consequence, the score, second variation, and Fisher information operator are unified through the cumulant structure of the process, placing likelihood-based inference for point processes on the same structural footing as in finite-dimensional models.
Daniel E. Clark (Fri,) studied this question.