[extropy-chat] three coins in a fountain, part 2: the bayesian angle
Eliezer S. Yudkowsky
sentience at pobox.com
Thu Oct 20 20:58:15 UTC 2005
> Several students might get all the answers right,
> but their scores would still differ: the cocksure
> Bayes-babe wins if she in that group, since she
> bet it all.
> Have you any suggestions for an overall strategy
> on such a test? Is there some systematic way for
> expressing your epsilon probabilities? You don't
> want to lose to the monkey, that would be
> embarrassing. Eliezer, how now?
Heh, nice try.
Probabilities exist in the mind, not in reality. There is no such thing
as the probability of a probability. You can assign probabilities over
long-run frequencies, probabilities over mixes of Everett branches,
probabilities over background physical parameters that you can only
approximate as producing 'random' frequencies, and - closest to what you
had in mind - the probability of a fixed computation producing an
output. For example, you could take a fixed program whose outputs are
intended to represent Bayesian computations of probability, and ask
after your subjective probability that the computation returns the
string "1/2" or "1/3".
But in the problem as you pose it, it is meaningless to ask whether the
coin's other side 'really' has a probability of tails of 1/3, or a
probability of tails of 1/2. The other side is either really tails, or
really heads. You can ask about the probability that the professor
thinks the probability is 1/3.
Suppose you have a probability of epsilon that the professor thinks the
answer is 1/2; or that an ideal computation calculating Bayesian
probabilities from the information given in some standardized format
would return the output "1/2". This doesn't mean your probability that
the coin is tails is more than 1/3. You could attach an equal and
balancing epsilon to the probability that an ideal computation returns
the answer "1/6". Thus the amount you bet on the actual coin's actual
other side being actually tails is still 1/3.
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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