How to practise Bayesian statistics outside the Bayesian church: What philosophy for Bayesian statistical modelling?

Unlike most other statistical frameworks, Bayesian statistical inference is wedded to a particular approach in the philosophy of science (see Howson & Urbach, 2006); this approach is called Bayesianism. Rather than being concerned with model fitting, this position in the philosophy of science primarily addresses theory choice. Naturally, in some cases there exists a relation between scientific theories and statistical models, and this relation can be so tight that choosing the model is tantamount to accepting the theory. However, in many cases of data analysis, the statistical model bears only an indirect relation to scientific theory, and in such cases the act of statistical modelling is distinct from the act of theory choice.