Generalised Bayesian Inference for Discrete Intractable Likelihood

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.离散状态空间（discrete state spaces）对统计推断构成了重大的计算挑战，这是因为归一化常数（normalisation constant）的计算需要对大型乃至可能无限的集合进行求和，该操作在实际中往往难以实现。本文针对该计算挑战，提出了一种适用于离散难处理似然（intractable likelihood）的新型广义贝叶斯推断（generalised Bayesian inference）方法。受近期连续数据方法论进展的启发，本文的核心思路是使用离散Fisher散度（Fisher divergence）替代存在问题的难处理似然，来更新关于模型参数的信念。该方法得到的广义后验（generalised posterior）可通过马尔可夫链蒙特卡洛（Markov chain Monte Carlo）等标准计算工具进行采样，从而规避了难以计算的归一化常数。本文分析了广义后验的统计性质，确立了后验一致性（posterior consistency）与渐近正态性（asymptotic normality）的充分条件。此外，本文还提出了一种用于广义后验校准的新型通用方法。本文针对离散空间数据的格点模型（lattice models）与计数数据的多变量模型（multivariate models）开展了应用研究，在两类场景中，该方法均能以较低的计算成本实现广义贝叶斯推断。