[extropy-chat] Re: monty hall paradox again: reds and green gorfs

[Eliezer Yudkowsky]

Spike wrote:
>> Eliezer Yudkowsky
>> 
>>> ...The probability that the other envelope is larger must go to 1/3,
>>> it doesn't matter how it gets there."
>> 
>> Your mathematician friend is flat wrong, and needs to study Bayesian 
>> probability theory. --  Eliezer S. Yudkowsky
> 
> OK cool, I was hoping someone would say that.  So where is the error?
> Are you saying that Bayesian reasoning predicts that there *is* a profit
> in swapping?  Even if you have no idea how much a zorg is?  How do you 
> set up the Bayesian priors?  How do you set up a sim to prove it?

The second part of the problem is how to deal with zorgs when "you have no
idea how much a zorg is worth".  Here's a question:  Do you think you are
just as likely to see an envelope containing 23,342,001,988 zorgs as you
are to see an envelope containing 12 zorgs?  Or to put it another way, do
you think that your chance of seeing a zorg amount between 1 and 10,000 is 
the same as your chance of seeing a zorg amount between 23,342,000,001 and
23,342,010,000?  Probably not, right?

Your mathematician friend is correct if, and only if, we use an evenly
distributed logarithmic prior for the value of a zorg - i.e., you think
that a zorg is equally likely to be worth an amount on the rough order of 1
dollar, 10 dollars, 1000 dollars, a hundredth of a cent, and so on.  So you
might open up the envelope and find 100 zorgs, then conclude that the total
amount being *exactly* 300 zorgs was twice as improbable (had half the
probability density) as the total amount being *exactly* 150 zorgs.  So the
probability of the other envelope being larger would go to 1/3.  But this
uniformly distributed logarithmic prior for the value of a zorg is improper 
for the same reason as the uniformly distributed smooth prior for the 
dollar amount; there has to be a cutoff somewhere, it is not exactly as 
likely that a zorg is worth 10^(1,556,823) dollars as one dollar.  For that 
matter, our scenario also assumes that the prior probability of any actual 
dollar amount in the envelopes is smoothly distributed between zero and 
infinity, another impropriety.

If we use a logarithmic prior from zero to infinity for the dollar amount, 
we always suppose (before looking) that the probability of the other 
envelope containing the larger amount is 1/2.  *After* looking, and 
discovering any specific amount such as $100, we always believe the other 
envelope has a 1/3 probability of containing $200, and a 2/3 probability of 
containing $50.  Magic?  No, an improperly formed infinite prior.  Pick any 
quantity 3T from an infinite continuous logarithmic prior, show a person 
either T or 2T at random, and he will always think it twice as likely that 
you showed him T as 2T.

Similarly, pick any envelopes T and 2T from an infinite smooth prior from 0 
to infinity, and your friend will always want to trade up.  I am reminded 
of Martin Gardner's proof that all numbers are tiny: no matter how large a 
finite number is, most numbers are very much larger.

Finally, your friend allegedly stated:

> What puzzles me even more is the reasoning of the gay mathematician, who
> had no problem with the idea that opening one envelope somehow causes
> the other one to become "probably the smaller one."  How?  I asked.  "It
> doesn't matter how," he calmly replied. "Physicists worry about that
> sort of thing, the crass empiricists.  Mathematicians do not."
> 
> I must have been wearing a stunned or puzzled countenance. "Well," he
> continued, "It must do it, right?  There cannot be any inherent profit
> to swapping.  You can write a quick sim to prove it.  The probability
> that the other envelope is larger must go to 1/3, it doesn't matter how
> it gets there."

1)  This is not a physics problem.  This is a Bayesian probability problem. 
  It is squarely the responsibility of mathematicians.

2)  There cannot be any inherent profit to swapping as a *universal, 
automatic* policy.  Once you look inside the envelope, you update your 
probabilities and may make an informed decision to swap or not swap.  The 
fundamental fallacy is that because I have an equal probability of giving 
you either envelope, you can look in an envelope and *then* say that the 
other envelope has an equal probability of being larger or smaller.  When 
you look in the envelope you must update your probabilities accordingly. 
This is why I say your mathematician friend is flat wrong, even though it 
is possible to construct an improper prior that makes him correct; the 
fundamental presumption that there must be no expected profit to trading, 
*after* you look inside the envelope, is un-Bayesian.  Bayesians are 
commanded (by E.T. Jaynes, that's who) to make full use of every tiny scrap 
of information.  If it makes us a profit, that is fine too, but the main 
thing is to be follow the perfect graceful math of Bayesian probability 
theory, as pure and bright as the white light of the full moon reflecting 
from a still pool of clear water.

-- 
Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence