Amara Graps says:
> James Wetterau (email@example.com) Tue, 26 Oct 1999 writes:
> >So far, the only non-circular definition I have found is a
> >mathematical definition claiming certainty in results iff we
> >take limits as the number of experiments goes to infinity.
> That's the "frequentist" definition of probability. I don't find the
> definition satisfying either. You may find the Bayesian perspective
> of probability more natural and intutive.
Thanks for the pointers! I'll check this Bayesian stuff out soon. I actually just encountered a discussion of Bayesian probability yesterday in an essay about psychological problems. It actually reminds me of an alternate model of probability that I've been struggling toward. This all started when my girlfirend came home from her job one day and challenged me to come up with a non-circular definition of probability and she was unsatisfied with the limit on the results of trials going to infinity.
To think aloud here the alternate conception I came up with was the idea of representing the proportional size of the unknown state sets that lead to certain results, as modified by the known factors. For a coin flip, for example, if I hold the coin at an identical position each time, but slightly increase the force each time I flip it, I suppose it should be possible to cause the coin to flip first n half rotations before landing, then n+1, up to n+m, at which point I can't impart more force. Now if I simply cannot accurately control the force, the outcome will be the result of the probability weighting of the range of forces that amount to each number of half flips. Of course there are lots of other factors in a typical coin flip, but what interests me in this imaginary controlled case is that even then I can see how the continuous range of states might lead to a fairly well balanced set of results, provided the state ranges are roughly equal in size.
What troubles me about this notion is that I seem to be saying that I can estimate a size of sets of unknowns based on actual outcomes. What does it really mean to say that the size of these sets have a certain proportion? There is some amount of force that will actually be imparted, so in a sense only one state point actually pertains. Perhaps it's time to do the reading. :-)
All the best,