Andrew Gelman Rejoinder Bayesian Analysis (2008) 3, Number 3, pp. 467-478.
Abstract. In the main article I presented a series of objections to Bayesian inference, written in the voice of a hypothetical anti-Bayesian statistician. Here I respond to these objections … .
Some suppose that Gelman promotes and justifies ‘Bayesianism’, the idea that Bayesian data analysis is adequate for all problems, with no need to ‘engage brain’. Equivalently, some suppose that Bayesian data analysis is only about the application of standard probability theory. But Gelman’s position is more nuanced. The emphasis in the following quotes is mine.
Bayesian inference (or, more generally, Bayesian data analysis) is a method for summarizing uncertainty and making estimates and predictions using probability statements conditional on observed data and an assumed model.
2 Responses to my own criticisms of Bayesian methods
As several discussants have noted, the objections to Bayesian methods in my original article were not entirely sincere. Or, to put it another way, these are sincere objections that I have thought through and, I believe, have largely been resolved.
The challenge comes in constructing realistic models and in assessing their fit.
… Bayesian inference (and also utility theory) are ideals or aspirations as much as they are descriptions.
I’ll retreat to the usual Bayesian answer that our default methods perform as well or better than classical default methods.
if information is available to distinguish the groups, then it can and should be added to the model.
Gelman is not advocating that Bayesian data analysis is always straightforward to apply or interpret. He does claim that its methods can be applied to important classes of problems, where they are at least as good – and, he argues, better than – conventional alternatives.
His last point is relevant to the interpretation of the results. For example, if a doctor tells you that you have a certain disease with a certain probability, this is probably based on some statistical study based on various groups. But in applying this result to you there is an implicit assumption that you are typical of the group. if you know that you come from a family or racial group where the incidence of the disease is markedly atypical (higher or lower), you would do well to mention this to your doctor.