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Alicorn comments on What is Bayesianism? - Less Wrong
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Eliezer's views as expressed in Blueberry's links touch on a key identifying characteristic of frequentism: the tendency to think of probabilities as inherent properties of objects. More concretely, a pure frequentist (a being as rare as a pure Bayesian) treats probabilities as proper only to outcomes of a repeatable random experiment. (The definition of such a thing is pretty tricky, of course.)
What does that mean for frequentist statistical inference? Well, it's forbidden to assign probabilities to anything that is deterministic in your model of reality. So you have estimators, which are functions of the random data and thus random themselves, and you assess how good they are for your purpose by looking at their sampling distributions. You have confidence interval procedures, the endpoints of which are random variables, and you assess the sampling probability that the interval contains the true value of the parameter (and the width of the interval, to avoid pathological intervals that have nothing to do with the data). You have statistical hypothesis testing, which categorizes a simple hypothesis as “rejected” or “not rejected” based on a procedure assessed in terms of the sampling probability of an error in the categorization. You have, basically, anything you can come up with, provided you justify it in terms of its sampling properties over infinitely repeated random experiments.
Here is a more general definition of "pure frequentism" (which includes frequentists such as Reichenbach):
Consider an assertion of probability of the form "This X has probability p of being a Y." A frequentist holds that this assertion is meaningful only if the following conditions are met:
The speaker has already specified a determinate set X of things that actually have or will exist, and this set contains "this X".
The speaker has already specified a determinate set Y containing all things that have been or will be Ys.
The assertion is true if the proportion of elements of X that are also in Y is precisely p.
A few remarks:
The assertion would mean something different if the speaker had specified different sets X and Y, even though X and Y aren't mentioned explicitly in the assertion.
If no such sets had been specified in the preceding discourse, the assertion by itself would be meaningless.
However, the speaker has complete freedom in what to take as the set X containing "this X", so long as X contains X. In particular, the other elements don't have to be exactly like X, or be generated by exactly the same repeatable procedure, or anything like that. There are practical constraints on X, though. For example, X should be an interesting set.
[ETA:] An important distinction between Bayesianism and Frequentism is this: Note that, according to the above, the correct probability has nothing to do with the state of knowledge of the speaker. Once the sets X and Y are determined, there is an objective fact of the matter regarding the proportion of things in X that are also in Y. The speaker is objectively right or wrong in asserting that this proportion is p, and that rightness or wrongness had nothing to do with what the speaker knew. It had only to do with the objective frequency of elements of Y among the elements of X.