[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

spike wrote:
> 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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