And your Bayesian statistician will acquire their priors how exactly?
Bayesian statistics can be very powerful, but it would be a terrible idea to prefer Bayesian approaches to Frequentist ones in all situations.
Bayesian statistics can be very powerful, but it would be a terrible idea to prefer Bayesian approaches to Frequentist ones in all situations.
As If Frequentists somehow didn't need priors. Everyone starts with prior knowledge. We might as well use it. Or do you advocate not using every scrap of knowledge available to you? That would be stupid.
Sure, prior knowledge can be shaky, or difficult to justify. But at least, a Bayesian will be explicit about it, instead of, like, sweeping normal probability distribution assumptions under the linear regression rug.
> it would be a terrible idea to prefer Bayesian approaches to Frequentist ones in all situations.
Name three examples that doesn't involve the Frequentist using better prior information than the Bayesian.
By the way, Bayesians know that using probability theory correctly is sometimes intractable (combinatorial explosion and all that). In those cases, they will use approximations. But at least, they will know it's an approximation.
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You really should read chapters 1 and 2 of Probability Theory: the Logic of Science. They give a good feel of why Bayesians are correct as a simple matter of fact.
They don't show that the frequentist approach is wrong. If both methods result in the same answer and frequentist methods are easier to use then what is the problem?
Also, the Frequentist approach is not wrong. It's inaccurate, to the extent its results differ from the Bayesian ones. This inaccuracy tend to go down as we gather more data. Which is a good thing, or else science itself wouldn't work.
One more thing. You said "in some sense". Are you seriously suggesting that the assumptions behind Cox's Theorem can reasonably be challenged?
Sure: probability is continuous.
Well… To me, it is obvious.
> There are both really good and horrible Frequentist and Bayesian statisticians.
Yeah. If I had to choose between Fisher and Anonymous Bayesian, I may chose Fisher.
However, unless both kind of statistics yield the same results (I don't think they do), then at least one of them is bogus, by application of the non-contradiction principle. So, while I can imagine there are good Frequentists Statisticians out there, I insist that frequentism itself is bogus.
I'm testing the effectiveness of a drug. Drugs of this class have a certain likelihood of working, the noise in my data is known, the experimental group did this much better than the control... does the drug really work? So far so trivial, in either Bayesianism or Frequentism. Now, I happen to mention that I tested 10000 variants of this drug and only sent data for the one that seemed to work. The rest aren't interesting after all. Under Frequentism, it's easy to take this into account. Under Bayesianism, it requires complex definitions of observations, and is easy to overlook as there's no space for it in the formula.
I have a collection of unfair dice. Unfortunately, they all look the same and got dumped on the floor. Now someone grabbed one off the floor at random and wants to make bets with me about it. Even experienced Bayesians are likely to mix up their propositions in a case like this. I say that from having read discussions of similar problems. Yes, if you do it right, it comes out correctly, but Frequentism makes sure you've thought about what you're asking in the same way Bayesianism makes sure you've thought about your priors.
Somebody else will have to give a third example.
Bayesianism and Frequentism are based on the same math, and math is math. If you use them correctly, they'll get you the same answer every time. The difference is what they make easy, and what mistakes they protect you against.
Second example: Okay, Bayesian statistics are harder. That's a disadvantage.
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> If you use [Bayesianism or Frequentism] correctly, they'll get you the same answer every time
Wat.
If they gave invariably the same answer, then, why the endless debates? By the way, here is an apparent factual disagreement bettwen Bayesianism and Frequentism:
There are very many real-world problems that have fast and accurate frequentist solutions, but slow and difficult Bayesian solutions. Despite my personal bias -- my research primarily relies on Bayesian inference -- I can't fathom how one can reasonably argue that frequentist approaches are always inferior, even in applied statistics.
My original claim is broader than I wanted it to be. The fact is, a Frequentist approach will always be less accurate than the correct application of probability theory. But of course,
> Bayesians know that using probability theory correctly is sometimes intractable (combinatorial explosion and all that). In those cases, they will use approximations. But at least, they will know it's an approximation.
https://news.ycombinator.com/item?id=6793905
The key to the Bayesian outlook is to remember that no matter what, there is a correct answer, even if you can't afford to compute it. As Eliezer Yudkowsky put it, there are laws of thought. Want to use Frequentist tools? Sure, why not. Just remember that they often violate the laws of ideal though. Some inaccuracy inevitably ensues.
https://www.andrew.cmu.edu/user/kk3n/simplicity/KassRaftery1...
(though one does have to specify the hypotheses to be tested more exactly than, say, "the effect is not zero.")