On 11/21/2022 9:37 PM, Bruce Kellett wrote:
On Tue, Nov 22, 2022 at 3:57 PM Jesse Mazer <[email protected]> wrote:
What about the idea of grounding the notion of probability in
terms of the frequency in the limit of a hypothetical infinite
series of trials, what philosophers call "hypothetical
frequentism"? The Stanford Encyclopedia of Philosophy discussion
of this at
https://plato.stanford.edu/entries/probability-interpret/#FreInt
notes the objection that the limit depends on the order we count
the trials, but it seems pretty natural to use temporal ordering
in this case. Aside from the philosophical objection that we don't
have any clear a priori justification for privileging temporal
ordering in this way, are there any objections of a more technical
nature to hypothetical frequentism with temporal ordering
(scenarios where it would give you a different answer from
standard probability theory), or are the objections purely
philosophical?
The standard trouble with the hypothetical infinite series of trials
is that we have to define the probability in terms of subsequences,
since we can't actually realize an infinite series. In order for these
subsequences to give (approximately) the same probability as the
hypothetical infinite series, the subsequences have to be "typical",
and "typical" can only be defined probabilistically, so we are back
with the problem of circularity.
Temporal ordering of the sequence is also somewhat arbitrary, since if
we order a series of coin tosses according to magnitude (heads = 0,
tails = 1), then most subsequences will not be "typical" and will give
spurious results. Temporal ordering implies that we have actually
completed an infinite series of tosses, and that is never possible. We
then have to assume that the first N trials form a "typical" subset,
and how do you ever justify that?
We did a lot statistical mechanics taking the limit of many
states/particles etc. without worrying that the experiments were never
infinite.
Brent
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