  Modern scientific cosmology pushes the boundaries of knowledge and the
knowable. This is prompting questions on the nature of scientific knowledge. A
central issue is what defines a 'good' model. When addressing global properties
of the Universe or its initial state this becomes a particularly pressing
issue. How to assess the probability of the Universe as a whole is empirically
ambiguous, since we can examine only part of a single realisation of the system
under investigation: at some point, data will run out. We review the basics of
applying Bayesian statistical explanation to the Universe as a whole. We argue
that a conventional Bayesian approach to model inference generally fails in
such circumstances, and cannot resolve, e.g., the so-called 'measure problem'
in inflationary cosmology. Implicit and non-empirical valuations inevitably
enter model assessment in these cases. This undermines the possibility to
perform Bayesian model comparison. One must therefore either stay silent, or
pursue a more general form of systematic and rational model assessment. We
outline a generalised axiological Bayesian model inference framework, based on
mathematical lattices. This extends inference based on empirical data
(evidence) to additionally consider the properties of model structure
(elegance) and model possibility space (beneficence). We propose this as a
natural and theoretically well-motivated framework for introducing an explicit,
rational approach to theoretical model prejudice and inference beyond data.
