# Generalised Bayesian Inference for Discrete Intractable Likelihood

arXiv: https://arxiv.org/abs/2206.08420 · HTML: https://mlsanity.com/paper/2206.08420/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 84 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 21 of 21)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [Blindness of score-based methods to isolated components and mixing proportions](https://mlsanity.com/paper/2008.10087/) (2020)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [Gradient Estimation with Discrete Stein Operators](https://mlsanity.com/paper/2202.09497/) (2022)
- [On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods](https://mlsanity.com/paper/1306.4032/) (2013)
- [General Bayesian Loss Function Selection and the use of Improper Models](https://mlsanity.com/paper/2106.01214/) (2021)
- [Objective Bayesian inference with proper scoring rules](https://mlsanity.com/paper/1711.10819/) (2017)
- [Bayesian model comparison with the Hyv"arinen score: computation and consistency](https://mlsanity.com/paper/1711.00136/) (2017)
- [A Review of Multivariate Distributions for Count Data Derived from the Poisson Distribution](https://mlsanity.com/paper/1609.00066/) (2016)
- [Nonparametric graphical model for counts](https://mlsanity.com/paper/1901.00886/) (2019)
- [Generalized Score Matching for Regression](https://mlsanity.com/paper/2203.09864/) (2022)
- [Physics-constrained Bayesian inference of state functions in classical density-functional theory](https://mlsanity.com/paper/2010.03374/) (2020)
- [Marginal Pseudo-Likelihood Learning of Markov Network structures](https://mlsanity.com/paper/1401.4988/) (2014)
- [Multivariate Conway-Maxwell-Poisson Distribution: Sarmanov Method and Doubly-Intractable Bayesian Inference](https://mlsanity.com/paper/2107.07561/) (2021)
- [Markov Chain Truncation for Doubly-Intractable Inference](https://mlsanity.com/paper/1610.05672/) (2016)

## Built on by (most cited first, 14 of 14)

- [Outlier-robust Kalman Filtering through Generalised Bayes](https://mlsanity.com/paper/2405.05646/) (2024)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Towards Healing the Blindness of Score Matching](https://mlsanity.com/paper/2209.07396/) (2022)
- [Robust and Conjugate Spatio-Temporal Gaussian Processes](https://mlsanity.com/paper/2502.02450/) (2025)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Robust Bayesian Optimisation with Unbounded Corruptions](https://mlsanity.com/paper/2511.15315/) (2025)
- [Generalised Robust Bayes for Joint Inference of Model and Contamination](https://mlsanity.com/paper/2607.25665/) (2026)
- [On the meaning of uncertainty for ethical AI: philosophy and practice](https://mlsanity.com/paper/2309.05529/) (2023)
- [Spatio-temporal probabilistic forecast using MMAF-guided learning](https://mlsanity.com/paper/2603.15055/) (2026)
- [Data Distribution Valuation Using Generalized Bayesian Inference](https://mlsanity.com/paper/2604.05993/) (2026)
- [Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data](https://mlsanity.com/paper/2609.01783/) (2026)

## Abstract (arXiv)

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.