# Re: Bayesian boxes and expectation value

From: Jacques M Mallah <jqm1584.domain.name.hidden>
Date: Fri, 21 May 1999 21:00:43 -0400

On Thu, 20 May 1999 GSLevy.domain.name.hidden wrote:
> MMM.... there is nothing better than an experiment to resolve the issue one
> way or another.

EXCUSE ME?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!
I think I have to refresh your memory a bit. Look at this quote
from a previous post:

You wrote:
>> Assumption 2)
>> It is common sense that it does not make any sense to switch.
>> Because of the PERFECT SYMMETRY of the situation

and I responded:
> Again, totally obvious and conventional, mentioned in the original
>post that started this.

Now you are somehow accusing me of advocating switching! I am
shocked and insulted!
It is totally obvious that with no additional information, there
is no advantage in switching. It is equally obvious that your
"explanation" for that fact, based on a completely mixed up formula for
expectation values, was totally WRONG. I was trying to explain to you why
it was wrong so that you could understand that.

> Certainly the simple minded expectation value of 0.5(2m + 0.5m) = 1.25m for
> switching does not make any sense. If you believe that it does make sense, I
> think I just have found a way to make a quick buck.

Again, that is the correct formula (for a logaritmic initial dist.),
but on average m is either zero or infinity so it's somewhat meaningless.
For decision making, however, it's natural to assume that m came from some
reasonable distribution and then one can show that there's no advantage to
switching, on average.
See http://www.artsci.wustl.edu/~chalmers/chalmers.envelope.html
for David Chalmers' analysis.

From: Gilles HENRI <Gilles.Henri.domain.name.hidden>
Subject: Re: Bayesian boxes and expectation value

>If you don't know the maximum value M, there is actually no rational for
>switching. The simple minded expectation value 0.5(2m + 0.5m) = 1.25m is
>not totally simple-minded: it evaluates correctly the expectation value of
>the ratio between what you would find in the second box to what you found
>in the first one. It is actually 1.25. However you should not use this
>expectation value to conclude that it is better to take the second box,
>because in the case where the second box contains 2m, it means also that
>the value of m that you found in the first box is lower than the average
>value 0.5*(m+2m) = 1.5m, and the opposite in the second case. The confusion
>arises from the fact that m is not an expectation value of anything, and
>summing 2m and 0.5m does not give an expectation value of what you can win,
>because of course the expectation value of the ratio is not the ratio of
>expectation values.
>That's better seen if you call m0 the (unknown) amount which is really put
>in the first box. Either you open the first box and you find m1=m0. The
>second one contains 2*m1=2*m0. Or you open the second one and you find
>m2=2*m0. The first one ontains 0.5*m2 = m0. The expectation value of the
>content of the first box you opened is <m> = 0.5*(m1+m2) = 0.5*(m0+2*m0)=
>1.5 m0. That of the other box is <m'> = 0.5*(2m1 + 0.5m2)=0.5*(2*m0+m0)=
>1.5 m0, which is the same of course. You see more clearly that the mistake
>comes from the fact that although m denotes always what you found in the
>first box, its expectation value is not the same following which box you
>assumed to have opened. However as I said, if you open systematically the
>second box,and you calculate each time the ratio between box 2 and box 1,
>you will indeed find an average value of 1.25. But it doesn't mean that you
>will gain more than if you kept systematically the first box, because the
>cases where the factor is 2 are just those where the value you found in the
>first box is lower...

Yes, that is basically correct.

From: GSLevy.domain.name.hidden
Subject: Re: Bayesian boxes and expectation value

>Gilles I fully agree with you. Your analysis is great.

That's nice. Gilles, I guess you succeeded in convincing him that
his earlier analysis was TOTALLY WRONG. Great.

- - - - - - -
Jacques Mallah (jqm1584.domain.name.hidden)
Graduate Student / Many Worlder / Devil's Advocate
"I know what no one else knows" - 'Runaway Train', Soul Asylum
My URL: http://pages.nyu.edu/~jqm1584/
Received on Fri May 21 1999 - 18:01:44 PDT

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