[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.

[kgwgk]

We agree that they don't always give the same answer then.

And also that the answers refer to different things. Bayesian methods depend on the prior probablity distribution chosen for the parameter of interest and produce a posterior probability distribution for it [1]. Frequentist methods have no concept of probability distribution regarding the parameter of interest at all.

So long!

[1] "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." I'm having issues to understand that though. If priors and posteriors are not "distributions over the quantity of interest" what are they about?

[whatshisface]

The posterior can be anything depending on the choice of prior. It is not a distribution over the variable of interest any more than a frequentist's ensembles are. Like frequentist answers, the posterior is understood to be something that can stand in for the in fact unobtainable distribution over the variable of interest.

[kgwgk]

> It is not a distribution over the variable of interest any more than a frequentist's ensembles are.

It can be right or wrong but it is what it is!

The frequentist ensembles in the frequentist inference are absolutely not a distribution over the variable of interest. They are defined for a fixed value of the variable of interest.

The 95% confidence interval means that 95% of the intervals that you generate in the ensemble where the paramater has always whatever its unknown actual value is will include it.

[whatshisface]

Wouldn't that imply that every Bayesian calculation ever done was wrong? The chances that you'd choose the right ultimate prior function R->R from that set are measure zero.

[kgwgk]

The calculation Input -> [Model] -> Output is not wrong.

I don't know what are you trying to say but it's no longer related to whether Bayesian methods and frequentist methods give the same answers. They don't even try to represent the same questions.

PS: I didn't have time to reply to your later comment, so here is a final comment reaching for common ground.

> frequentist conceptual machinery

Just to be clear, that's not what I've been talking about. As I'm tried to make clear several times I'm talking about what is commonly understood as frequentist methods. I gave the concrete example of confidence intervals. I started the conversation with a explicit question: What do you call "frequentist methods"?

I agree that if we define "frequentist conceptual machinery" to be "probability" we can do Bayes with it.

[whatshisface]

Frequentism and bayesianism are more than separate collections of formulas, they're philosophical stances on interpreting theorems about statistics. That's where the separate language comes from. It might not be taught very often (because it's inconvenient and wordy), but it is actually possible to address anything in frequentist language, even cases where you're combining separate sources of differently-weighted evidence. The same goes for bayesian language.