Prasanta Bandyopadhyay and Gordon Brittan write:
We introduce a distinction, unnoticed in the literature, between four varieties of objective Bayesianism. What we call ‘strong objective Bayesianism’ is characterized by two claims, that all scientific inference is ‘logical’ and that, given the same background information two agents will ascribe a unique probability to their priors. We think that neither of these claims can be sustained; in this sense, they are ‘dogmatic’. The first fails to recognize that some scientific inference, in particular that concerning evidential relations, is not (in the appropriate sense) logical, the second fails to provide a non-question-begging account of ‘same background information’. We urge that a suitably objective Bayesian account of scientific inference does not require either of the claims. Finally, we argue that Bayesianism needs to be fine-grained in the same way that Bayesians fine-grain their beliefs.
I have not read their paper in detail but I think I pretty much agree with their criticism of classical or strong Bayesian philosophies of the objective or subjective variety. In particular, I agree with them that (a) the traditional Bayesian philosophy (which culminates in the posterior probability of a model being true) is not a good model for the evaluation and replacement of scientific theories, but (b) a fuller, falsificationist Bayesian philosophy can do the job.
I’d just like to add a few remarks:
1. As always, I find it misleading to focus on the prior distribution as the locus of subjective uncertainty. The data model is just as subjective. Or, I should say, it depends on context. In some problems, there is more reasonable agreement on the population model; in others, there is more agreement on the data model. It’s just that, for historical reason, “likelihood methods” have been grandfathered in as classical methods and thus don’t suffer the Bayesian taint.
It’s kind of like the Bible. All sorts of goofy stories that happened to have been placed in time before 100 BC become canonical; whereas everything that happened after is evaluated in the category of “history” rather than “religion.” This cuts both ways: in one direction, you have people who believe anything that happens to be in the official collection of biblical stories; on the other, historical stories get the benefit of being revisable by evidence. Something similar happens in many statistical problems, when all sorts of critical thinking gets applied to the prior distribution, whereas conventional likelihoods just get accepted.
Before going on to the next issue, let me qualify the above by recognizing Deborah Mayo’s point that, in typical cases, the data model differs from the prior distribution by being more accessible to checking. In practice, though, statisticians (including those of the classical or Bayesian variety who complain or rejoice about the subjectivity of prior distributions) often don’t take the opportunity to check the fit of their data models.
2. I don’t like the example on page 50. In the problems I’ve worked on, it’s never seemed to make any sense to talk about the posterior probability that a continuous parameter equals zero or that a particular model is true. As I’ve written on various occasions, I can see how such procedures can be useful but I don’t see them making any logical sense.
3. Statistical reasoning often seems to lend itself to a two-level expression of belief. For example, in evaluating a research paper, a reviewer might express some uncertainty about whether a result is truly statistically significant. This sort of logic seems odd from a scientific perspective (it’s sort of like evaluating the weight of an object by assessing how heavy it looks), but, in the context of the sociology of science, the evaluation of evidence is clearly an important thing that we do.
As I learned from Thomas Basbøll, Plato characterizes knowledge as “justified, true belief.” I like this definition, and it gives a clue as to the scientific relevance of statements such as, “I don’t think this finding is actually statistically significant,” even in completely Bayesian settings.
4. Bandyopadhyay and Brittan write, “Personalist Bayesians like Bruno de Finetti and Leonard J. Savage claim that there is only one such condition, coherence.” I’ve written about this before, but let me briefly say again that I see coherence as a structuring property rather than an attribute of inference. In nontrivial settings, our inferences won’t be coherent—if they were, we could just skip Bayesian inference, posterior integration, Stan, etc., and simply look at data and write down our subjective posterior distributions. When our inferences aren’t close to coherent, though, this is a problem. Thus, I think coherence is a valuable concept, but not because Bayesian inferences are coherent (they’re not) but because Bayesian inference provides a mechanism for finding and resolving incoherences.
5. Finally, I’ll link to my five papers on the philosophy of Bayesian inference:
Philosophy and the practice of Bayesian statistics (with Cosma Shalizi)
Philosophy and the practice of Bayesian statistics in the social sciences (with Cosma Shalizi)