I don't see why you need the subsequences to be "typical" if you are using temporal ordering--for example if you want to define the probability of heads, you can define f_N as the fraction of flips that came up heads in the first N trials of the temporal ordering, then consider the limit as N approaches infinity. Any specific value of N may be highly atypical while the infinite limit is still 1/2. So again, is the objection just the philosophical one that temporal ordering seems "arbitrary", and/or some kind of philosophical objection to defining probability in terms of a complete infinite series of trials even though this is explicitly a "hypothetical" definition?
On Tue, Nov 22, 2022 at 12:37 AM Bruce Kellett <[email protected]> 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? > > Bruce > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CAFxXSLQSx91DMcoNZfcGVWwJ_gMBd1pv2Xk%3DgBYQ3eij_owcvg%40mail.gmail.com > <https://groups.google.com/d/msgid/everything-list/CAFxXSLQSx91DMcoNZfcGVWwJ_gMBd1pv2Xk%3DgBYQ3eij_owcvg%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAPCWU3K4Yj%3DFQd8C%3DckrOXqjMTq7%2B3dhQBLMhPpJ4KdiRyBysQ%40mail.gmail.com.
