Re: poly: What probabilities?

Robin Hanson writes:
> If you find your estimate to be very senstive
> conditioning on each new peice of information you consider when trying
> to construct your estimate, you might want to guess at a distribution
> over what your estimate will be if your thought about it for another
> ten minutes. That is a legit probability over probabilities.

Yes, I can see that there would be situations where your internal
probability estimate is converging too slowly to come up with a meaningful
final figure.

My wife suggested another case, where experimental results are being
expressed in probabilistic terms (like, the probability of a 50 year
old woman dying of breast cancer is X%, and increases to Y% if her
mother had the disease). In that case there may be error bars from the
underlying statistics which were used to derive the probabilities.

> Note that none of this requires any revision of the standard
> probability picture.

In the standard mathematical theory of probability, probability is defined
as a measure function over a set of disjoint outcomes, such that the sum
of the measures of all the outcomes is 1. Can the idea of "probability
of probability" be formalized? Do all probabilities have a probability?

My son (age 15) suggested that a probability of 100% should always itself
have a probability of 100%. It would not make sense, he thought, to say
that there was a 90% probability that the probability of an event is 100%.
Either an event is certain or it is not, and you should not say that it
is certain unless you are certain that it is certain. Does that make
sense?

I am still not completely comfortable with these second order
probabilities. It makes me wonder why we stop there.

Hal
Received on Mon Apr 27 02:45:48 1998