[whatshisface]

>In a situation where you only have one source of information, Bayesian and Frequentist methods usually give the same answer.

Bayesian and frequentist methods always give the same answer because they represent two different ways of translating the same mathematical ideas into English.

[kgwgk]

What do you call "frequentist methods"?

Imagine the following question: "what's the male/female ratio in gorillas?"

A frequentist method may provide the answer "[1.1 1.3] is a 95% confidence interval" based on taking a sample of zoos and asking them about the sex of the last gorilla born there.

A Bayesian method will provide a different answer - maybe one difficult to reconcile. Because it's not "translating the same mathematical ideas into English". Not only the translation is different - the "mathematical ideas" considered are different as well.

[whatshisface]

I don't think that's a complete case, you can't just say that the bayesian answer would be difficult to reconcile without offering it.

[kgwgk]

A Bayesian may put a strong prior on around the 1:1 sex ratio at birth - because in addition to that data regarding a sample of births they incorporate in the calculation knowledge about the plausible ratio coming from previous observations or biological facts about giraffes or related animals - and get a 95% credible interval (which is conceptually completely different from a 95% confidence interval) like [0.99 1.01] or whatever.

You can't just say that Bayesian and frequentist methods _always_ give the same answer without offering even a _single_ example.

What is commonly understood as 'Bayesian methods' will give answers in the form of a probability distribution. What is commonly understood as 'frequentist methods' will never do that. How can they always give the same answer then?

[whatshisface]

Thanks for adding detail. I didn't offer any examples because I want to use ones that sound representative to you.

The 95% confidence interval is in reference to a probability distribution, I'm not sure what you mean when you say that frequentist answers aren't in terms of those.

As for your bayesian answer, there is a prior that would make their result equal to the frequentist one - and in another example where priors were more obviously crucial (weak evidence) the frequentist would still use Baye's theorem. Their ensemble would be all possible worlds in which they'd ask the question.

[kgwgk]

> The 95% confidence interval is in reference to a probability distribution, I'm not sure what you mean when you say that frequentist answers aren't in terms of those.

Maybe I should have been more explicit. Let me complete what I wrote above:

"What is commonly understood as 'Bayesian methods' will give answers in the form of a probability distribution FOR THE QUANTITY OF INTEREST. What is commonly understood as 'frequentist methods' will never do that."

In the sex ratio example, the 95% confidence interval [1.1 1.3] DOES NOT mean that the probability that the sex ratio is between 1.1 and 1.3 is 95%. What it means is that when you calculate confidence intervals using this method 95% of them will contain the true value of the parameter. It's not the same thing.

> As for your bayesian answer, there is a prior that would make their result equal to the frequentist one

So it doesn't _always_ give the same answer, does it? (In the best case you get the same answer by twisting what a confidence interval means and only if the "magic" prior is used.)

If you mean that frequentist methods shouldn't be used if they give a different answer I agree!

The problems with frequentist methods have been known for decades: https://www.jstor.org/stable/2281869

They are used because the are easy even though they give the wrong answer (or the answer to the wrong question) - not because they always give the right answer.

[whatshisface]

Two bayesians can come up with different answers if they use different priors, there are rarely clear-cut rules for choice of prior (if there is a commonly accepted value, how much weight do you put on it?). Two frequentists can come up with different answers if they define their ensembles differently. A frequentist and a bayesian can come up with the same answer if the implicit prior built in to the ensemble matches the explicit prior adopted by the bayesian.

Since bayesians understand that they can't trace back their probability updates all the way back to a single ultimate prior, they do not actually talk about distributions over the quantity of interest either. Ultimately it's the same as what a frequentist would do.

When a bayesian lets a somewhat arbitrarily adopted prior stand in for the "ultimate prior," so that they can interpret their answer as a distribution over the variable of interest, they are not doing anything differently from when a frequentist fudges the exact concepts and treats their answer about the degree to which the model would predict the measured outcomes as if it was the probability of the model being true.

There are no schools of statistics that offer an unqualified "probability that the model is true," and that may be philosophically impossible AFAIK.