Abstract

Modern Bayesian statistics and machine learning tools are being applied to increasingly complex physical and biological phenomenon, and as a result make use of increasingly complex models. One such class of models are so-called “doubly intractable” models, for which the likelihood function is known only up to normalisation constant. Examples are as varied as models of multivariate count data arising in genomics, lattice models arising in statistical physics, or even large protein signalling network models arising in biochemistry. Unfortunately, the Bayesian treatment of such problems presents two main challenges. Firstly, the size of these models and lack of tractability of the likelihood creates significant computational challenges, with standard MCMC or variational methods not directly applicable. Secondly, the complexity of the underlying phenomena means that the models proposed by scientists are often partly incomplete, and as a result misspecified. To solve these issues, we propose a novel class of generalised Bayesian posteriors, which depart from the classical Bayesian approach by updating beliefs through loss functions instead of likelihoods. We will show how this approach allows us to select loss functions which provide both computational tractability and robustness to misspecification, and illustrate the approach on examples in genomics, physics and biochemistry which are beyond the scope of current techniques.