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/
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