  Motivated by big data and the vast parameter spaces in modern machine
learning models, optimisation approaches to Bayesian inference have seen a
surge in popularity in recent years. In this paper, we address the connection
between the popular new methods termed generalised Bayesian inference and Bayes
linear methods. We propose a further generalisation to Bayesian inference that
unifies these and other recent approaches by considering the Bayesian inference
problem as one of finding the closest point in a particular solution space to a
data generating process, where these notions differ depending on user-specified
geometries and foundational belief systems. Motivated by this framework, we
propose a generalisation to Bayes linear approaches that enables fast and
principled inferences that obey the coherence requirements implied by domain
restrictions on random quantities. We demonstrate the efficacy of generalised
Bayes linear inference on a number of examples, including monotonic regression
and inference for spatial counts. This paper is accompanied by an R package
available at github.com/astfalckl/bayeslinear.
