Robust Generalised Bayesian Inference for Intractable Likelihoods
This work addresses robustness in Bayesian inference for statisticians and practitioners dealing with intractable likelihoods, such as in exponential family or graphical models, though it is incremental as it builds on existing generalised Bayesian methods.
The paper tackled robust Bayesian inference for models with intractable likelihoods by using a Stein discrepancy as a loss function, circumventing normalisation constants and achieving closed-form or MCMC-accessible posteriors, with theoretical guarantees like consistency and asymptotic normality demonstrated in numerical experiments on intractable distributions.
Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Carlo. On a theoretical level, we show consistency, asymptotic normality, and bias-robustness of the generalised posterior, highlighting how these properties are impacted by the choice of Stein discrepancy. Then, we provide numerical experiments on a range of intractable distributions, including applications to kernel-based exponential family models and non-Gaussian graphical models.