Against overly restrictive definitions: No, I don’t think it helps to describe Bayes as “the analysis of subjective
 beliefs” (nor, for that matter, does it help to characterize the statements of Krugman or Mankiw as not being “economics”)

I get frustrated when people use overly restrictive definitions of something they don’t like.

Here’s an example of an overly restrictive definition that got me thinking about all this. Larry Wasserman writes (as reported by Deborah Mayo):

I wish people were clearer about what Bayes is/is not and what 
frequentist inference is/is not. Bayes is the analysis of subjective
 beliefs but provides no frequency guarantees. Frequentist inference 
is about making procedures that have frequency guarantees but makes no 
pretense of representing anyone’s beliefs.

I’ll accept Larry’s definition of frequentist inference. But as for his definition of Bayesian inference: No no no no no. The probabilities we use in our Bayesian inference are not subjective, or, they’re no more subjective than the logistic regressions and normal distributions and Poisson distributions and so forth that fill up all the textbooks on frequentist inference. See chapter 1 of BDA for lots of examples of prior distributions that are objectively assigned from data. Here’s my definition of “Bayesian.” Science in general has both subjective and objective aspects. Science is always full of subjective human choices, and it’s always about studying larger questions that have an objective reality.

Now, don’t get me wrong—there are lots of good reasons for wanting to avoid the use of prior distributions or to use various non-Bayesian methods in different applications. Larry writes, “In our
 world of high-dimensional, complex models I can’t see how anyone can 
interpret the output of a Bayesian analysis in any meaningful way,” and I have no doubt of his sincerity. I myself have difficulty interpreting the output of non-Bayesian analyses in the high-dimensional, complex models that I work on—I honestly find it difficult to think about non-Bayesian estimates of public opinion in population subgroups, or a non-Bayesian estimate of the concentration of a drug in a complex pharmacology model—but I accept that Larry’s comfort zone is different from mine, and I think it makes a lot of sense for him to continue working using methods that he feels comfortable with. (See here for more of this sort of talk.) So, it’s fine with me for Larry to report his discomfort with Bayesian inference in his experience. But please please please don’t define it for us! That doesn’t help at all.

To get back to Larry’s definition: Yes, “the analysis of subjective beliefs” is one model for Bayes. You could also label classical (or frequentist) statistics as “the analysis of simple random samples.” Both definitions are limiting. Yes, Bayes can be expressed in terms of subjective beliefs, but it can also be applied to other settings that have nothing to do with beliefs (except to the extent that all scientific inquiries are ultimately about what is believed about the world). Similarly, classical methods can be applied to all sorts of problems that do not involve random sampling. It’s all about extending the mathematical model to a larger class of problems.

I do think, by the way, that these sorts of interactions can be helpful. I don’t agree with Larry’s characterization of Bayesian inference, but, conditional on him believing that, I’m glad he wrote it down, because it gives me the opportunity to express my disagreement. Sometimes this sort of exchange can help. For example, between 2008 and 2012 Larry updated his believes regarding the relation of randomization and Bayes. In 2008: “randomized experiments . . . don’t really have a place in Bayesian inference.” In 2013: “Some people say that there is no role for randomization in Bayesian inference. . . . But this is not really true.” This is progress! And I say this not as any kind of “gotcha.” I am sincerely happy that, through discussion, we have moved forward. Larry is an influential researcher and explicator of statistics, and I am glad that he has a clearer view of the relation between randomization and Bayes. From the other direction, many of my own attitudes on statistics have changed over the years (here’s one example).

In case this helps, here are some of my thoughts on statistical pragmatism from a couple years ago.

And, just to be clear, I don’t think Larry is being personally aggressive or intentionally misleading in how he characterizes Bayesian statistics. He’s worked in the same department as Jay Kadane for many years, and I think Jay sees Bayesian statistics as being all about subjective beliefs. And once you get an idea in your head it can be hard to dislodge it. But I do think the definition is aggressive, in that it serves to implicitly diminish Bayesian statistics. Once you accept the definition (and it is natural for a reader to do so, as the definition is presented in a neutral, innocuous manner), it’s hard to move forward in a sensible way on the topic.

P.S. Larry also writes of “Paul
 Krugman’s socialist bullshit parading as economics.” That’s another example of defining away the problem. I think I’d prefer to let Paul Krugman (or, on the other side, Greg Mankiw) define his approach. For better or worse, I think it’s ridiculous to describe what Krugman (or Mankiw) does as X “parading as economics,” for any X. Sorry, but what Krugman and Mankiw do is economics. They’re leading economists, and if you don’t like what they do, fine, but that just means there’s some aspect of economics that you don’t like. It’s silly to restrict “economics” to just the stuff you like. Just to shift sideways for a moment, I hate the so-called Fisher randomization test, and I also can’t stand the inverse-gamma (0.001, 0.001) prior distribution—but I recognize that these are part of statistics. They’re just statistical methods that I don’t like. For good reasons. I’m not saying that my dislike of these methods (or Larry’s dislike of Krugman’s economics) is merely a matter of taste—we have good reasons for our attitudes—but, no, we don’t have the authority to rule that a topic is not part of economics, or not part of statistics, just because we don’t like it.

Oddly enough, I don’t really have a problem with someone describing Krugman’s or Mankiw’s writing as “bullshit” (even though I don’t personally agree with this characterization, at least not most of the time) as with the attempt to define it away by saying it is “parading as economics.” Krugman’s and Mankiw’s writing may be bullshit, but it definitely is economics. No parading about that.

P.P.S. The above post is from 2014. When it appeared, Jordan Ellenberg commented that I should consider posting it once a year. I haven’t done that, but it’s been six years now, so I thought it was worth posting again. Also, in the years since the appearance of the above post, Christian Hennig and I wrote our paper, Beyond subjective and objective in statistics.