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 constants）的计算需要对庞大乃至无限的集合进行求和，这在实际中往往难以实现。本文针对这一计算难题，提出了一种适用于不可处理似然（intractable likelihood）的新型广义贝叶斯推断（generalised Bayesian inference）方法。该方法受近期连续数据领域方法论进展的启发，核心思路是使用离散费希尔散度（discrete Fisher divergence）替代存在计算难题的不可处理似然，来更新关于模型参数的信念。由此得到的广义后验分布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，从而规避了不可处理的归一化常数的计算。本文分析了广义后验分布的统计性质，推导得到了后验一致性与渐近正态性的充分条件。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以及计数数据的多变量模型开展了应用研究，结果表明该方法可在较低计算成本下实现广义贝叶斯推断。