Objective Bayes is Objective

I came across the following short comment from Larry Wasserman:

Larry Wasserman. “Frequentist Bayes is objective (comment on articles by Berger and by Goldstein).” Bayesian Anal. 1 (3) 451 - 456, September 2006. https://doi.org/10.1214/06-BA116H

In this, Wasserman agrees that objective Bayesianism is, well, objective. (At least insofar as “objective” is a useful description at all in statistics. [1])

I interpret “Objective Bayesian inference” to mean “Bayesian methods that have good frequency properties.” I refer to these methods as Frequentist-Bayes…

While I disagree with this change in nomenclature, namely because Objective Bayesianism (in the Williamsonian sense) is more than just calibrated Bayes and frequentist theory doesn’t own Calibration, it places Objective Bayesianism squarely in sight.

Interestingly, he goes on to remind us that frequentist insistence on good coverage, a common thump on the subjectivist Bayesian, is (sometimes) not enough.

[Y]ou can have procedures with good coverage that sometimes yield unacceptable results, such as an empty confidence set. This does not invalidate coverage as a desirable property; it merely shows that coverage is a weak property. The solution is to demand more than just coverage, not to throw out coverage.

When coverage is not enough, we should ask for more properties in addition to coverage rather than abandoning coverage.

(emphasis mine)

Coverage is necessary but not sufficient. I’m not suggesting that we choose any old method with correct coverage and declare victory; I’m suggesting this is a minimal requirement.

This is great to hear. The Objective Bayesian can agree with the frequentist that empirical calibration is an important norm, but not the only norm for inference. Wasserman discusses a context in which certain risk-sensitive intervals are to be preferred over mere coverage. I tend to agree that, as is common with norms generally, there is a degree of context sensitivity to their application – some contexts have more stringent requirements than others. Myself, I (always tentatively) think that Probabilism applies much more generally than Calibration, which applies more broadly than Equivocation, and that there are likely many more domain- and problem-appropriate norms that go beyond this, justified as the situation arises. Unlike Wasserman here, I think that there are perhaps contexts in which Calibration is not an operative norm (e. g. philosophical), though I suspect it would be reasonable for some to say that perhaps those contexts would not qualify as “statistical” anymore even though I, myself, don’t accept such a dichotomy.

Perhaps the most interesting thing here to me is what is not said. After all, a calibrated Bayesianism is not a frequentist theory. It changes the interpretations of fundamental quantities in frequentism (for the better!) like the confidence interval [2] and the ability to talk about the probability of hypotheses. These are not minor! Finally answering the right question is an achievement of sorts for a theory, and the Objective Bayesian can welcome new Bayes-Freqs anytime.

References

[1] Gelman, A., & Hennig, C. (2017). Beyond subjective and objective in statistics. Journal of the Royal Statistical Society. Series A (Statistics in Society), 180(4), 967–1033. http://www.jstor.org/stable/44682661

[2] Williamson, J. Why Frequentists and Bayesians Need Each Other. Erkenn 78, 293–318 (2013). https://doi.org/10.1007/s10670-011-9317-8