[extropy-chat] three coins in a fountain, part 2: the bayesian angle

[spike]

> bounces at lists.extropy.org] On Behalf Of spike
> Subject: [extropy-chat] three coins in a fountain
> 
> You have a bag containing three coins: a double headed,
> a double tailed and an ordinary coin.  You reach into
> the bag without looking, take one out and place it on
> the table.  You see a head facing up.
> 
> What is the probability that the other side is a tail?...
> 
> spike


Suppose you are taking a class in Bayesian reasoning.  

Your professor decides your grade based on competition 
with the other students: a third of the class will get 
an A, a third gets a B, a third a C.  

Your grade is based on a take-home multiple choice test,
ten questions like the coins example above, four 
possible responses to each.  The trick is this: instead
of adding up the number of correct responses in the
normal (boring) way, on this test you will assign a
probability or confidence level to each possible
response.  Then your score will be determined by
taking the *product* of your assigned probability to
each correct answer.

For instance, if the three coins example above had
the following choices:

a) 1/2
b) 1/3
c) 0
d) something other than these


You immediately know that choice c) is wrong (clearly
there is some possibility of a tail) so that answer
gets p(c)=0.  You are pretty sure the answer is b) 1/3,
but you could be wrong; there is an admittedly
suspect line of reasoning that leads to choice (a). 
So you assign the probabilities p(b)=0.7 and p(a)=0.3,
so p(d)=0.

Now you recognize that if you are wrong and somehow
choice (d) is correct, then you get 0 for that question,
so your overall score on the test is a mighty goose egg,
since your final score is the product of your assigned
probabilities on each correct answer.  

It wouldn't cost your final score much to give 
choice (d) some kind of epsilon, say .001 just 
as insurance against the zonk, right?

The class is graded as a competition with the other 
students, and you know the self-confident hot-ass
Bayesian theorist will assign p=1 to each of her
responses.  If she is right on all, she scores an
impressive 1, bur if she misses even one, she gets 
a goose egg.  The timid probability non-grokker
always-loses-at-poker guy will not even read the
test, but rather assigns p=.25 to everything, then
answers randomly, for a score of (.25)^10 or about
1e-6.  A monkey could do that well, but he might
beat the Baysian grokmeister if she goofs even one.
Otherwise probably last.

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?

spike