Glen, Okay, given some of the later postings against the original question, I am thinking that your question may have morphed or that I have completely misunderstood what you are asking. Not sure. For example, somehow we have gone from probability theory and its ontological status to the Banach-Tarski Theorem and the Axiom of Choice. This seems like a non-sequitur, but not sure. First off, a theory is inductive, whereas, a theorem is deductive; so that is my first disconnect. So I don't understand how we got here ... but this often happens to me. :-(
Then we go to what I think is a refinement of the original question. Yes? (I am just trying to navigate the thinking to get to the core issue, that I seem to be missing): But what is this "set of events"? That's the question that is being > discussed on this thread. It turns out that the events for a finite space > is nothing more than the set of all possible combinations of the sample > points. (Formally the event set is something called a "sigma algebra", but > no matter.) So, an event scan be thought of simply all combinations of the > sample points. and then to: So, the events already have probabilities by virtue of just being in a > probability space. They don't have to be "selected", "chosen" or any such > thing. They "just sit there" and have probabilities - all of them. The > notion of time is never mentioned or required. An event is not *all* the combinations of the sample points. As Grant has said, an event [outcome] has probability depending on how it is arbitrarily configured from the event space by the researcher. Moreover, there is an important distinction to be made between the distribution of values [e.g., the numbers on each side of a dice being equally likely] and the sampling distribution that is dependent on how the event is composed in a trial sequence. The sampling distribution is the mathematical result of the convolution of probabilities when choosing N independent, *usually *identically-distributed random picks from the parent distribution. Another example might be helpful: I think you are trying to define the sample space like with an urn of 10 balls with three red and seven white. An event, in that case, would be something like picking three balls all red. We could easily compute the probability of this event by using hypergeometric arithmetic; this is because of the sample space changing if you do not replace any balls after each pick. But, there is a finite number of other possible events in this scenario of picking three things from a bin of ten things. To be sure, though, this statistical problem does not relate at all to the paradoxical Axiom of Choice ... unless I am still missing something. We are not interested in slicing and dicing [no pun intended] a probability space of a certain size in a way for coming up with, say, two identical but mutually exclusive probability spaces of the same size. This would make no sense, IMHO. Events are just the outcome(s) one is interested in computing the probability for. They don't exist--as selections, in the way that I think you mean--until they are formulated by the researcher ... not trying to conjure up anything spooky here between the observer and the experiment as at the quantum level. :-) Nor are these events--not being mathematical entities of any type--something to be discovered in some platonic math sense [I mistakenly called you a Platonist, but on rereading the thread, I think you are not. Sorry. But the world wouldn't be as interesting without Platonists. :-) ]. For example, there is the possible event of being dealt four aces in one hand of five cards and for which I can assign a probability given the conceptual structure of the probability space: a deck of cards. This is nothing more than laying out the number of possible [combinations--so order doesn't matter] of hands (a sample) and determining how many ways I could be dealt four aces [just one] ... then dividing the latter by the former. This is an example of a categorical probability space, where the events are all the various ways [combinations] one can be dealt five cards from a deck of 52. We could go on to define these into categories like two of a kind, three of a kind, and so forth. Each of those events can be then assigned a probability. and then: Perhaps it's helpful to think about the "axiom of choice"? Is a > "choosable" element somehow distinct from a "chosen" element? Does the act > of choosing change the element in some way I'm unaware of? Does > choosability require an agent exist and (eventually) _do_ the choosing? The Axiom of Choice is a paradox that seems to get into trouble with set-cardinality, where it comes to infinite sets. To me is nothing more than a mathematical curiosity that has no impact on the practical world. So I don't think this is helpful to your cause. But I would be more than curious to see how you think it might be. I am more an applied mathematician|statistician than anything like a theoretical mathematician; though, I have happily worked with many of the latter ... and hopefully the reverse was true. :-) Okay, back to your observation: the fact that it is possible to choose a particular event from the set of all possible events in the event space is a trivial requirement. I cannot, for example, pick a black ball--an impossible event--from the previous urn of only red and white balls. So being able to choose three red balls from that urn makes the event "choosable." Is that event then distinct from that same event that has been "chosen?" At the classical level--as opposed to the quantum level--I cannot see any meaningful distinction EXCEPT to say that the former event is a possibility and the second event is a realization ... and that the way such events get discussed in practical probability and statistics. There is no spooky agent that needs to get factored into the calculus, IMHO. Somehow, I still feel I am missing something. Maybe you can figure it out, but it may not be all that important, and your question may have already been addressed satisfactorily by the other responses posted to the thread. Cheers On Wed, Dec 14, 2016 at 2:41 PM, Frank Wimberly <wimber...@gmail.com> wrote: > Don't think about choosing. The axiom of choice says that there is a > function from each set (subset) to an element of itself, as I recall. > > Frank > > > Frank C. Wimberly > 140 Calle Ojo Feliz > Santa Fe, NM 87505 > > wimber...@gmail.com wimbe...@cal.berkeley.edu > Phone: (505) 995-8715 Cell: (505) 670-9918 > > -----Original Message----- > From: Friam [mailto:friam-boun...@redfish.com] On Behalf Of glen ? > Sent: Wednesday, December 14, 2016 11:36 AM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] probability vs. statistics (was Re: Model of > induction) > > > Ha! Yay! Yes, now I feel like we're discussing the radicality > (radicalness?) of Platonic math ... and how weird mathematicians sound (to > me) when they say we're discovering theorems rather than constructing them. > 8^) > > Perhaps it's helpful to think about the "axiom of choice"? Is a > "choosable" element somehow distinct from a "chosen" element? Does the act > of choosing change the element in some way I'm unaware of? Does > choosability require an agent exist and (eventually) _do_ the choosing? > > > > On 12/14/2016 10:24 AM, Eric Charles wrote: > > Ack! Well... I guess now we're in the muck of what the heck probability > and statistics are for mathematicians vs. scientists. Of note, my > understanding is that statistics was a field for at least a few decades > before it was specified in a formal enough way to be invited into the > hallows of mathematics departments, and that it is still frequently viewed > with suspicion there. > > > > Glen states: /We talk of "selecting" or "choosing" subsets or elements > > from larger sets. But such "selection" isn't an action in time. Such > > "selection" is an already extant property of that organization of > > sets./ > > > > I find such talk quite baffling. When I talk about selecting or choosing > or assigning, I am talking about an action in time. Often I'm talking about > an action that I personally performed. "You are in condition A. You are in > condition B. You are in condition A." etc. Maybe I flip a coin when you > walk into my lab room, maybe I pre-generated some random numbers, maybe I > look at the second hand of my watch as soon as you walk in, maybe I write > down a number "arbitrarily", etc. At any rate, you are not in a condition > before I put you in one, and whatever it is I want to measure about you > hasn't happened yet. > > > > I fully admit that we can model the system without reference to time, > > if we want to. Such efforts might yield keen insights. If Glen had > > said that we can usefully model what we are interested in as an > > organized set with such-and-such properties, and time no where to be > > found, that might seem pretty reasonable. But that would be a formal > > model produced for specific purposes, not the actual phenomenon of > > interest. Everything interesting that we want to describe as > > "probable" and all the conclusions we want to come to "statistically" > > are, for the lab scientist, time dependent phenomena. (I assert.) > > -- > ☣ glen > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe > http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove >
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
