The roles of inductive modelling and coherence in Bayesian statistics

Summary

The role of the inductive modelling process (IMP) seems to be of practical importance in Bayesian statistics; it is recommended that the statistician should emphasise meaningful real-life considerations rather than more formal aspects such as the axioms of coherence. It is argued that whilst axiomatics provide some motivation for the Bayesian philosophy, the real strength of Bayesianism lies in its practical advantages and in its plausible representation of real-life processes. A number of standard procedures, e.g., validation of results, choosing between different models, predictive distributions, the linear model, sufficiency, tail area behaviour of sampling distributions, and hierarchical models are reconsidered in the light of the IMP philosophy, with a variety of conclusions. For example, whilst mathematical theory and Bayesian methodology are thought to prove invaluable techniques at many local points in a statician’s IMP, a global theoretical solution might restrict the statistician’s inductive thought processes. The linear statistical model is open to improvement in a number of medical and socio-economic situations; a simple Bayesian alternative related to logistic discrimination analysis often, leads to better conclusions for the inductive modeller.