My problem is not just with the methods—although I do have problems with the method—but also with the ideology.
My problem with the method
It’s the usual story. Bayesian inference is model-based. Your model will never be perfect, and if you push hard you can find the weak points and magnify them until you get ridiculous inferences.
One example we’ve talked about a lot is the simple case of the estimate,
theta_hat ~ normal(theta, 1)
that’s one standard error away from zero:
theta_hat = 1.
Put a flat prior on theta and you end up with an 84% posterior probability that theta is greater than 0. Step back a bit, and it’s saying that you’ll offer 5-to-1 odds after seeing an observation that is statistically indistinguishable from noise.
That was easy. More complicated examples will have more complicated problems, but the way probability works is that you can always find some chink in the model and exploit it to result in a clearly bad prediction.
What about non-Bayesian methods: they’re based on models too, so they’ll also have problems? For sure. But Bayesisan inference can be worse because it is so open: you can get the posterior probability for anything.
Don’t get me wrong. I still think Bayesian methods are great, and I think the proclivity of Bayesian inferences to tend toward the ridiculous is just fine—as long as we’re willing to take such poor predictions as a reason to improve our models. But Bayesian inference can lead us astray, and we’re better statisticians if we realize that.
My problem with the ideology
As the saying goes, the problem with Bayes is the Bayesians. It’s the whole religion thing, the people who say that Bayesian reasoning is just rational thinking, or that rational thinking is necessarily Bayesian, the people who refuse to check their models because subjectivity, the people who try to talk you into using a “reference prior” because objectivity. Bayesian inference is a tool. It solves some problems but not all, and I’m exhausted by the ideology of the Bayes-evangelists.
Tomorrow: What’s wrong with null hypothesis significance testing.