Rejoinder to discussion of ‘Philosophy and the practice of Bayesian statistics’

The main point of our paper was to dispute the commonly held view that Bayesian statistics is or should be an algorithmic, inductive process culminating in the calculation of the posterior probabilities of competing models. Instead, we argued that effective data analysis – Bayesian or otherwise – proceeds more messily through a jagged process of formulating research hypotheses, exploring their implications in light of data, and rejecting aspects of our models in light of systematic misfits compared to available data or other sources of information. We associate this last bit with Popper’s falsification or (weakly) with Kuhn’s scientific revolutions, but these connections with classical philosophy of science are not crucial. Our real point is that Bayesian data analysis, in the form that we understand and practise, requires the active involvement of the researcher in constructing and criticizingmodels, and that from this perspective the entire process of Bayesian prior-to-posterior inference can be seen as an elaborate way of understanding the implications of a model so that it can be effectively tested. Just as a chicken is said to be nothing but an egg’s way of making another egg, so posterior inference is a way that a model can evaluate itself. But this inference and evaluation process, in our view, has essentially nothing to do with calculations of the posterior probability of competingmodels. Aswe discuss in our paper, for technical, philosophical, and historical reasons we tend not to trust such marginal posterior probabilities. (See Figure 1 of our paper for the sort of reasoning that we do not like.) Given all this, the discussion of our paper is remarkably uncontentious: none of the five discussants express support for the standard (according to Wikipedia) view of an overarching inductive Bayesian inference, and all agree with us that the messiness of