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Tyrrell_McAllister comments on What is Bayesianism? - Less Wrong

81 Post author: Kaj_Sotala 26 February 2010 07:43AM

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Comment author: Tyrrell_McAllister 26 February 2010 09:19:32PM *  7 points [-]

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:

  1. The speaker has already specified a determinate set X of things that actually have or will exist, and this set contains "this X".

  2. 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:

  1. 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.

  2. If no such sets had been specified in the preceding discourse, the assertion by itself would be meaningless.

  3. 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.

  4. [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.