[extropy-chat] Coin Flip Paradox
The Avantguardian
avantguardian2020 at yahoo.com
Tue Jan 30 23:51:23 UTC 2007
--- gts <gts_2000 at yahoo.com> wrote:
> > There is great beauty in principles such as the
> Principle of
> > Indifference...
>
> Here is another example of where this "beautiful
> principle" fails
> miserably:
>
> This is from a text on decision theory [1], in which
> the author rejects
> the principle as a decision-making rule on two
> grounds:
>
> 1) it is illogical and philosophically untenable (if
> we have no reason to
> expect one outcome more than another then we have no
> reason to assume they
> are equiprobable, either)
>
> and, perhaps more critically,
>
> 2) the principle can lead to disaster...
>
> A decision must be made between action 1 (A1) and
> action 2 (A2). One of
> two scenarios will unfold (S1 or S2). We are
> ignorant of the probabilities
> of S1 and S2, so we invoke the principle of
> indifference (also called the
> principle of insufficient reason, as in this text)
> and assign each
> scenario a probability of 50%.
>
> As below, the expected utility (EU) of A1 is
> therefore 100, calculated as
> (.5 * -100) + (.5 * 300) = 100. The expected utility
> of A2 is 20,
> similarly calculated.
>
> S1 S2 EU
> ----------------------------------
> A1| -100 300 100
> A2| 10 30 20
>
> We choose A1 as this action offers the highest
> expected utility.
>
> The author writes, "The principle of insufficient
> reason could lead us to
> disaster... If, unbeknownst to us, the probability
> of S1 were, say, .9,
> the expected utility of A1 would be significantly
> less than that of A2. In
> a life or death situation the principle could be
> totally disastrous."
The principle of indifference is not a subtitute for
real information on the probability of the outcomes,
IF any such information is available. If you do have
any sort of informative prior, you are better off
using it than an uninformative prior every time. That
is almost the first thing they teach you in Bayesian
reasoning, use whatever data you have, and only when
you have zero data do you use the principle of
insufficient reason.
That Bayes law cannot predict outcomes reliably in an
informational vaccuum is not at all a weakness of the
principle. It just tells you the importance of
information in making decisions. How would a
frequentist decide between A1 and A2 in your above
game of life and death? By dying a million times until
they had a good estimate of the probability of S1?
As a matter of fact if the probability of S1 is .9
then the entire game has an overall expected utility
of -48. Which means it is a game not worth playing no
matter what option you choose.
Stuart LaForge
alt email: stuart"AT"ucla.edu
"If we all did the things we are capable of doing, we would literally astound ourselves." - Thomas Edison
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