  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.
