Eric, (I am ending many sentences with prepositions; apologies.)
Modern language usage manuals, for example,* Garner's Modern American Usage* [2009: 3rd Edition, page 654], advise that you no longer have to worry about ending a sentence with a preposition. As Winston Churchill once quipped when criticized for occasionally ending a sentence with a preposition, "This is the type of errant pedantry up with which I will not put." 🤐 Cheers, Robert W. On Mon, Dec 12, 2016 at 3:21 PM, Robert J. Cordingley <rob...@cirrillian.com > wrote: > Hi Eric > > I was remembering that if you tossed a perfectly balanced coin and got 10 > or 100 heads in a row it says absolutely nothing about the future coin > tosses nor undermines the initial condition of a perfectly balanced coin. > Bayesian or not the next head has a 50:50 probability of occurring. If you > saw a player get a long winning streak would you really place your bet in > the same way on the next spin? I would need to see lots of long runs (data > points) to make a choice on which tables to focus my efforts and we can > then employ Bayesian or formal statistics to the problem. > > I think your excellent analysis was founded on 'relative wins' which is > fine by me in identifying a winning wheel, as against 'the longer a run of > success' finding one which I'd consider very 'dodgy'. > > Thanks Robert > > > > On 12/12/16 1:56 PM, Eric Smith wrote: > >> Hi Robert, >> >> I worry about mixing technical and informal claims, and making it hard >> for people with different backgrounds to track which level the conversation >> is operating at. >> >> You said: >> >> A long run is itself a data point and the premise in red (below) is false. >>> >> and the premise in red (I am not using an RTF sender) from Nick was: >> >> But the longer a run of success continues, the greater is the probability >>>> that the wheel that produces those successes is biased. >>>> >>> Whether or not it is false actually depends on what “probability” one >> means to be referring to. (I am ending many sentences with prepositions; >> apologies.) >> >> It is hard to say that any “probability” inherently is “the” probability >> that the wheel produces those successes. A wheel is just a wheel (Freud or >> no Freud); to assign it a probability requires choosing a set and measure >> within which to embed it, and that always involves other assumptions by >> whoever is making the assertion. >> >> Under typical usages, yes, there could be some kind of “a priori” (or, in >> Bayesian-inference language, “prior”) probability that the wheel has a >> property, and yes, that probability would not be changed by testing how >> many wins it produces. >> >> On the other hand, the Bayesian posterior probability, obtained from the >> prior (however arrived-at) and the likelihood function, would indeed put >> greater weight on the wheel that is loaded, (under yet more assumptions of >> independence etc. to account for Roger’s comment that long runs are not the >> only possible signature of loading, and your own comments as well), the >> more wins one had seen from it relatively. >> >> I _assume_ that this intuition for how one updates Bayesian posteriors is >> behind Nick’s common-language premise that “the longer a run of success >> continues, the greater is the probability that the wheel that produces >> those successes is biased”. That would certainly have been what I meant in >> a short-hand for the more laborious Bayesian formula. >> >> >> For completeness, the Bayesian way of choosing a meaning for >> probabilities updated by observations is the following. >> >> Assume two random variables, M and D, which take values respectively >> standing for a Model or hypothesis, and an observed-value or Datum. So: >> hypothesis: this wheel and not that one is loaded. datum: this wheel has >> produced relatively more wins. >> >> Then, by some means, commit to what probability you assign to each value >> of M before you make an observation. Call it P(M). This is your Bayesian >> prior (for whether or not a certain wheel is loaded). Maybe you admit the >> possibility that some wheel is loaded because you have heard it said, and >> maybe you even assume that precisely one wheel in the house is loaded, only >> you don’t know which one. Lots of forms could be adopted. >> >> Next, we assume a true, physical property of the wheel is the probability >> distribution with which it produces wins, given whether it is or is not >> loaded. Notation is P(D|M). This is called the _likelihood function_ for >> data given a model. >> >> The Bayes construction is to say that the structure of unconditioned and >> conditioned probabilites requires that the same joint probability be >> arrivable-at in either of two ways: >> P(D,M) = P(D|M)P(M) = P(M|D)P(D). >> >> We have had to introduce a new “conditioned” probability, called the >> Bayesian Posterior, P(M|D), which treats the model as if it depended on the >> data. But this is just chopping a joint space of models and data two ways, >> and we are always allowed to do that. The unconditioned probability for >> data values, P(D), is usually expressed as the sum of P(D|M)P(M) over all >> values that M can take. That is the probability to see that datum any way >> it can be produced, if the prior describes that world correctly. In any >> case, if the prior P(M) was the best you can do, then P(D) is the best you >> can produce from it within this system. >> >> Bayesian updating says we can consistently assign this posterior >> probability as: P(M|D) = P(D|M) P(M) / P(D). >> >> P(M|D) obeys the axioms of a probability, and so is eligible to be the >> referent of Nick’s informal claim, and it would have the property he >> asserts, relative to P(M). >> >> Of course, none of this ensures that any of these probabilities is >> empirically accurate; that requires efforts at calibrating your whole >> system. Cosma Shalizi and Andrew Gelman have some lovely write-up of this >> somewhere, which should be easy enough to find (about standard fallacies in >> use of Bayesian updating, and what one can do to avoid committing them >> naively). Nonetheless, Bayesian updating does have many very desirable >> properties of converging on consistent answers in the limit of long >> observations, and making you less sensitive to mistakes in your original >> premises (at least under many circumstances, inluding roulette wheels) than >> you were originally. >> >> To my mind, none of this grants probabilities from God, which then end >> discussions. (So no buying into “objective Bayesianism”.) What this all >> does, in the best of worlds, is force us to speak in complete sentences >> about what assumptions we are willing to live with to get somewhere in >> reasoning. >> >> All best, >> >> Eric >> >> >> On Dec 12, 2016, at 12:44 PM, Robert J. Cordingley <rob...@cirrillian.com> >>> wrote: >>> >>> Based on https://plato.stanford.edu/entries/peirce/#dia - it looks like >>> abduction (AAA-2) to me - ie developing an educated guess as to which might >>> be the winning wheel. Enough funds should find it with some degree of >>> certainty but that may be a different question and should use different >>> statistics because the 'longest run' is a poor metric compared to say net >>> winnings or average rate of winning. A long run is itself a data point and >>> the premise in red (below) is false. >>> >>> Waiting for wisdom to kick in. R >>> >>> PS FWIW the article does not contain the phrase 'scientific induction' R >>> >>> >>> On 12/12/16 12:31 AM, Nick Thompson wrote: >>> >>>> Dear Wise Persons, >>>> Would the following work? >>>> Imagine you enter a casino that has a thousand roulette tables. The >>>> rumor circulates around the casino that one of the wheels is loaded. So, >>>> you call up a thousand of your friends and you all work together to find >>>> the loaded wheel. Why, because if you use your knowledge to play that >>>> wheel you will make a LOT of money. Now the problem you all face, of >>>> course, is that a run of successes is not an infallible sign of a loaded >>>> wheel. In fact, given randomness, it is assured that with a thousand >>>> players playing a thousand wheels as fast as they can, there will be random >>>> long runs of successes. But the longer a run of success continues, the >>>> greater is the probability that the wheel that produces those successes is >>>> biased. So, your team of players would be paid, on this account, for >>>> beginning to focus its play on those wheels with the longest runs. >>>> FWIW, this, I think, is Peirce’s model of scientific induction. >>>> Nick >>>> Nicholas S. Thompson >>>> Emeritus Professor of Psychology and Biology >>>> Clark University >>>> http://home.earthlink.net/~nickthompson/naturaldesigns/ >>>> >>>> >>>> ============================================================ >>>> 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 >>>> >>> -- >>> Cirrillian >>> Web Design & Development >>> Santa Fe, NM >>> >>> http://cirrillian.com >>> >>> 281-989-6272 (cell) >>> Member Design Corps of Santa Fe >>> >>> ============================================================ >>> 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 >> >> >> > -- > Cirrillian > Web Design & Development > Santa Fe, NM > http://cirrillian.com > 281-989-6272 (cell) > Member Design Corps of Santa Fe > > > ============================================================ > 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 >
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