Whilst successful in my career and user of probability, statistics, and inference on a regular basis, I simply cannot understand what's being discussed here.
I don't even want to understand it. Just like quantum, half the argument seems to be the a mismatch between mental models and actual reality.
Every time I've tried to understand the entire argument it just raises more questions to me. For example as I was first introduced to it, frequentists simple count frequencies observed in nature and then compute stats on them, and then build inferential models using those stats without assuming any complex underlying distribution. While Bayesians count frequencies, apply a prior correction (say, adding a pseudocount of one for every unobserved possible event, or any other way of assuming the generative process has a distribution that we've previously estimated), some stats,then build models from that.
however, after I was told that, I've seen several other arguments that quickly dive into: the distribution of the underlying events (I've heard that frequentists assume one type while bayesian assume another). Other folks just sort of give the example of the base rate fallacy.
Throughout all of this I've realized: I don't understand stats at all. I came to the scientific world with a view much more like physics: there is a microscopic event system (a particle simulation, or whatever) that we are observing, but due to limitations, we can only make macroscopic observations, which represent biased aggregations of the underlying microscopic event system. We can figure out those biases and use the aggregate data to build predictive models of the underlying systems- without ever really knowing the true details of the microscopic model.
From what I can tell, everything about what physicists do to model the world mentally is more Bayesian than Frequentist, if I understand what the hell people mean when they argue about it. However, as I said, every time I look at the arguments, I realize I don't understand stats, while I understand the physics approach which seems to be fairly obvious.
> For example as I was first introduced to it, frequentists simple count frequencies [...]. While Bayesians count frequencies [...]
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.
Did you read Wasserman's article? There's a difference between a frequentist/Bayesian interpretation of probability and a frequentist/Bayesian method of inference. I don't have to take a position on what probability "really means" to use either kind of inference method. (As the article says, the true difference has a lot more to do with wanting guaranteed coverage...)
If it's just a perspective on how to analyze probabilities, the physicists already knew how to derive equations for macroscopic observables and combinatorics, in ways that correspond to reality as we understand it. Thanks for the long writeup, but you basicalyl just confirmed I wasn't missing anything fundamental.
I don't even want to understand it. Just like quantum, half the argument seems to be the a mismatch between mental models and actual reality.