Bayesian continued and shuffling

Continuation of thread "bayes" 9/24.
Seems everyone kill filed it missing my questions.

Extremely interesting. May we dumb this down a little and
make it more poker relevant?

The best way I can think of to show I'm understanding
is last night during a game for the entire time the table
was intact someone probably had a pocket pair under.
We stopped counting after 20 in the first 1/2 hour.

As a player I had to assume the cards were in such an
order that our group was shuffling them in such a way
that pocket pairs would keep coming (which they did)
and I adjusted my reads and betting style accordingly.
From a Bayesian point of view did I not use prior knowledge?

We've all had games where a card or two continually
shows itself at relevant times and we adjust play. Bayesian?

What about bingo where a ball pops up almost every game
and people find cards with that number on it..

An observation about hand shuffling:
Please don't merge this statement with machine shufflers
they are different animals. or are they?

Try these shuffle experiments.
1) Do your normal shuffle with cut card in the deck. note where
the cut card is. Keep shuffling, keep noting placement.

2) Pick two cards that are next to each other. Shuffle. Where are
the two cards in relation to each other now. How about after
some other number of shuffles.

In general, for home games and my bar league we generally
do the ruffle 3-4 times and deal. Some people just do a few cut
type shuffles. (Can I say) obviously, the cards do not get totally
randomized and will tend to stay grouped and get grouped since people
keep certain cards. These get collected together with inadequate
shuffling. Seems to me these build up of actions is going to create a
valid prior history predictable by Bayesian theorem. Wrong?

For the record I try not to adjust and keep using pure probability
when playing but sometimes you get that streak. In poker terms we
call it a rush. I myself am learning to identify when I'm on a rush
and play cards I wouldn't normally play. I do this based on the above
shuffle statement.

Should I be?
How should I be using Bayesian theory while at the table?
Aside from tells what do you consider to be a valid priors.

AJ


Following is quote:

RichD wrote:
Every so often I see a reference to Bayes'
theorem somewhere; it's usually like "scientists
in biology/economics/etc. are breaking new
ground, by application of Bayesian analysis.
They are using Bayes' theorem, which is really
advanced and profound and everything..."

(( snip ))
Can anyone explain to me what the big deal
is on this formula,

(( snip ))

The triumphal hoop-la that one encounters from time to time is on
account of the fact that so-called Bayesian statistical inference is
*not* universally accepted as the answer to the problem of statistical
inference. There is also so-called classical statistical inference,
which is more entrenched, and generally more accepted for purposes of
scientific reporting. The Bayesian's have been laying siege to the
classical fortress for quite some time, and every little victory is
celebrated.

The problem lies not is Bayes theorem per se, which is, precisely, a
theorem in the theory of probability. The problem lies in how Bayes
theorem is applied to the problem of statistical inference. In this
regard, I am not sure that Rev. Thomas Bayes is the originator of this
theorem, but the literature certainly gives him credit for
investigating
the idea that the probability law governing the data, f(x; theta) --
where x is the data, theta some parameter, and f the probability
function -- could be construed as a conditional probability,
f(x|theta),
with x and theta in effect conceived as covarying in probabilistic
terms.

If that conception is accepted, the application of Bayes theorem then
allows us to assert, conceptually:

Posterior = Likelihood x Prior

where the Prior, b(theta), say, is the representation of uncertainty
regarding theta prior to the experimental data under consideration,
the
Likelihood is L(theta) = f(x|theta), and the Posterior is b(theta|x),
and where I leave out normalization details, taken care of in the
theorem, needed to ensure that the Posterior satisfies the condition
required of a probability distribution that it integrate to unity.

This is problematic for many reasons, none of them having to do with
Bayes theorem per se, and all of them having to do with the semantics
of
the inferential set-up. The debate has gone on, apparently unresolved,
from the late 18-th century down to the present. The dispositive
argument, for me, against the Bayesian construct, is that the problem
of
inference is to characterize what *the data* say about the unknown
parameter of interest, or perhaps some function transformation
thereof,
and perhaps in the presence of nuisance parameters. In this statement
of
the problem, the Prior is in-principle irrelevant. (If there is
relevant
prior data at hand, it could in principle be combined with the data.)
It
is the functional equivalent of arguing from facts not in evidence,
which as we all should know is not allowed in court, and for good and
valid reason. It is also why a Bayesian analysis would not do for
reporting scientific results. Even if an argument could be made that
prior data, or prior beliefs *should* form part of the inferential
procedure (in decision applications for example), the in-principle
question would remain of how to draw inference from what the data, and
only the data -- however construed, and including or not including
"prior" data -- say.

Be that as it may, it is interesting to note that the Bayesian schema
reduces to

Posterior = Likelihood

whenever a flat form of Prior is applied, as is often done when there
is
no basis on which to assert any sort of Prior (so-called ignorance
Prior). In that case, all that is achieved by Bayes theorem is to take
the Likelihood function and transmute it into probability density.

It is my contention that a true method of inference must find rules
for
manipulating the Likelihood function -- which alone is the entire
encapsulation of what the data say regarding the parameter of
interest,
under the assumed probability model -- that permit at minimum (i) the
elimination of nuisance parameters, and (ii) the extension of
uncertainty, consistent with the Likelihood, to function
transformations
of the parameter (for example loss functions). Bayesian triumphs
merely
derive from the fact that the Posterior is for the most part merely
the
Likelihood rescaled, unless some invincible Prior is used that
overwhelms all evidence of the data. In other words, the Bayesian
analysis is more or less "over the target", therefore it should be
expected that some bombs will hit. Nevertheless, the proper calculus
that should be associated with Likelihood is a possibility calculus,
not
a probability calculus. In the absence of any Prior, if the
Likelihood,
as a function of theta, were considered on its known, it would be
clear
that it is not probability density. It does however, to quote Fisher,
provide a "natural order for the possibilities under consideration."
In
other words, the Likelihood is a form of possibility distribution.

The insights of fuzzy set theory allow the two forms of uncertainty --
possibilistic and probabilistic -- to be intimately mixed. In the
asymptotic case (plenty of data), the results obtained will be the
same
regardless of the methods applied. But, where data are few, there will
be differences, and the Bayesian approach, in my opinion, will be
misleading, and not only because of the unwarranted inclusion of data
not in evidence. The rules for extension of possibilistic uncertainty
under function transformations are also different from the
probabilistic
rules. I have addressed these issues in my _Fuzziness and Probability_
(1995).

I hope this is helpful.

Regards,
S. F, Thomas
.

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