Posted by
Nick Thompson on
Dec 13, 2016; 2:26am
URL: http://friam.383.s1.nabble.com/Model-of-induction-tp7588431p7588445.html
Robert,
I want to get back to you eventually concerning what kind of -duction we are
talking about here.
But before that< I want to clear up any other confusions I may have.
Let's take your coin example; it's all my poor civilian brain can really
handle.
You are quite right that if coin is a balanced coin, a run of 100 heads is
no reason to believe that the next flip will be a heads. On the other hand,
after 100 heads, what is the probability that the coin is balanced? I used
to play a game in my freshman class in which I would bring in "my own
special" coin, and flip it for them. After each flip, I would ask them
whether they still believed that the coin was fair. What amazed me was with
what consistency people fell off the wagon between .10 and .05. The would
help to answer the question, later on. Why psychologists tended to use the
.05 level of significance.
N
Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/-----Original Message-----
From: Friam [mailto:
[hidden email]] On Behalf Of Robert J.
Cordingley
Sent: Monday, December 12, 2016 3:21 PM
To: The Friday Morning Applied Complexity Coffee Group <
[hidden email]>
Subject: Re: [FRIAM] Model of induction
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
<
[hidden email]> 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.
--
Cirrillian
Web Design & Development
Santa Fe, NM
http://cirrillian.com281-989-6272 (cell)
Member Design Corps of Santa Fe
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