This equation is science in a nutshell. On the left-hand-side it has the Bayesian Evidence. This is the quantity which is responsible for updating our belief in a scientific model in light of data (for example, how much we believe in the concordance model of cosmology in light of new supernovae measurements). On the right hand side it has the posterior averaged loglikelihood, and the Kullback Liebler divergence.
The first of these is a Bayesian “goodness of fit” measure, and the second of these is an information theory “model complexity”. The equation therefore quantifies Occam’s razor, telling us that the degree to which a model is scientifically convincing is composed in equal part how well it describes the data and in the simplicity with which it does so.
This equation and philosophical interpretation was discovered as part of a DiRAC supported Bayesian study in cosmology [2102.11511, equation 9].
The paper showcases some of the subtle but important details relevant when applying Bayesian model comparison in a modern cosmological setting. As concrete examples it considers the challenge of using measurement of the Universe to weigh and determine the hierarchy of neutrino masses, as well as detecting primordial gravitational waves generated near to the big bang during cosmic inflation. Such tasks are theoretically, observationally and computationally extremely demanding, requiring high performance computing to extract robust inferences from cosmological data. The paper emphasises that as we move to more and more stringent constraints on the tensor-to-scalar ratio using BICEP and LiteBIRD these considerations will become increasingly vital for interpreting the scientific content of increasingly powerful experiments.