Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper addresses this computational challenge through the development of a novel generalised Bayesian inference procedure suitable for discrete intractable likelihood. Inspired by recent methodological advances for continuous data, the main idea is to update beliefs about model parameters using a discrete Fisher divergence, in lieu of the problematic intractable likelihood. The result is a generalised posterior that can be sampled from using standard computational tools, such as Markov chain Monte Carlo, circumventing the intractable normalising constant. The statistical properties of the generalised posterior are analysed, with sufficient conditions for posterior consistency and asymptotic normality established. In addition, a novel and general approach to calibration of generalised posteriors is proposed. Applications are presented on lattice models for discrete spatial data and on multivariate models for count data, where in each case the methodology facilitates generalised Bayesian inference at low computational cost.

翻译：离散状态空间对统计推断构成了重大计算挑战，因为归一化常数的计算需要对大规模或可能无限的集合进行求和，这在实际中往往不可行。本文通过开发一种适用于离散不可计算似然的新型广义贝叶斯推断程序，旨在应对这一计算挑战。受连续数据领域最新方法论进展的启发，其主要思想是使用离散Fisher散度替代棘手的不可计算似然，来更新对模型参数的信念。由此产生的广义后验分布可通过标准计算工具（如马尔可夫链蒙特卡洛方法）进行采样，从而规避不可计算的归一化常数。本文分析了广义后验分布的统计特性，建立了后验一致性与渐近正态性的充分条件。此外，还提出了一种新颖且通用的广义后验校准方法。本文在离散空间数据的格点模型与计数数据的多元模型上展示了应用实例，在这两类模型中，该方法均能以较低的计算成本实现广义贝叶斯推断。