What Is Bayesian/Frequentist Inference? (2012)

I think that it is a bad way to explain differences. The good way is to look into the history of approaches and to see how they are different.

The history is illuminating. Frequentists started with card games, trying to figure out a winning strategy. And so they were attracted to frequencies, they invented combinatorics to calculate frequencies, and later they came with game theory. Of course it is not the whole story. While initially they get frequencies as given or inferred with math, they also encountered problems where it was impossible to calculate frequencies by combinatorics, so they invented a limit with samples approaching infinity of an empiric frequency, claimed it a definition of a probability, and now they deal with the impracticality of an infinity, using p-values or whatever to measure should they get more samples or it is enough already.

Thomas Bayes came from the other side. He started with a task where he had a hypotheses and tried to choose between them based on evidence. He was a priest and he was unsure should we believe in miracles given reports of eyewitnesses. So he dealt with a belief. He quantified belief and found a procedure of updating belief given a piece of evidence.

So generally speaking, Thomas Bayes started with the problem which frequentists saw as a side issue. Frequentists sought how to use probabilities to win a game without bothering much where to get those frequencies, Bayes sought how to infer a belief (or probabilities as a degrees of a belief) from an evidence, without bothering much what to do with the resulting belief. (To stop being a priest? I don't know what his plan was and I suspect he had no plan, it was a pure curiosity.).

And hence comes the ideological difference between them. Frequentists see probability as a property of a Universe, Bayesians see probability as a property of an observer, a property of his imperfect model of a Universe. Bayesians bring model explicitly into a picture, and so they can consciously think of enhancing it. Frequentists can think of a model too, but they lack vocabulary, it is a missing part of their picture, it is hard for them to pinpoint it.

It really has something in common with quantum mechanics that debated for decades is uncertainty a way the Universe works or it is just our imperfect way to describe it.

But these are ideological differences. To see practical differences one needs to dive into practical problems and to see how different approaches works there. Mostly people learn frequentist's approach in an undergraduate course and then they learn bayesianism on a bunch of problems that are easy with bayesianism and very hard or impossible to tackle with frequencies. You can try "Think Bayes"[1] if you like. Or to read Judea Pearl's "The Book of Why"[2]. He invented modern bayesianism, starting with ideas of Thomas Bayes. "The Book of Why" more of his next invention (Causation) but he talks there of bayesianism too.

[1] https://greenteapress.com/wp/think-bayes/

[2] https://www.amazon.com/Book-Why-Science-Cause-Effect-ebook/d...

Frequentists are fundamentally unable to answer "Given that I drew a cookie with 2 chips on the first draw, what is the chance I draw a cookie with 0 chips on my second draw?". That question requires a prior; any solution frequentists find is just implicitly assigning a prior to the distribution of jars.

You might claim that we should do away with intervals and just report posteriors for all physical quantities. This complicates matters slightly without solving the problem. If the true parameters, when ultimately known, consistently end up in very low density regions of the probability distribution (such as far in the tails), we would regard our uncertainty estimates as poor. Again, it is not hard to construct examples where Bayesian methods have poor coverage properties in this sense.

(Also, a minor point: Regarding "Do you need to select a strategy up front which gives at least a 70% chance of being right about the jar over iterations where the jar is fixed but the data varies?", one does not need to fix the jar for frequentist methods to have good guarantees. See Wasserman's simulation with the median.)

Re: "Given that I drew a cookie with 2 chips on the first draw, what is the chance I draw a cookie with 0 chips on my second draw?", this is not a question about estimating an unknown parameter of a distribution, so it's not statistical in the sense Wasserman is talking about. It's just an elementary probability question that requires knowing something about the jars to answer. Both frequentist and Bayesian statisticians agree on the validity of Bayes rule (and hence how to answer this question once the relevant information is known or assumed); where they differ is on how to conceptualize and estimate unknown parameters of probability distributions.