Hello Rich,
This'll have to be a very quick response as I am off on holiday after tomorrow
and have lots to get thru. You asked for a reply, so I try and state my position
once more. But agree that this is maybe not so useful to discuss yet again.
My claim is (provisionally, I don't wish to be dogmatic) that a probability is
best viewed as a property of the world, and not of a person.
For example, I play a game with a friend involving bets on balls in a
bag. I'm no good at judgement, and also a rotten cheat. So I dial up my
friend on my mobile, and whilst he is out of the room dealing with the phone
I quickly empty the bag of balls and count ten reds in a bag of fifty red and black.
Is that a property of me or the ball bag? (I won the game)
Let's look at something more subjective; where it is harder to take a population
sample. I was thinking about this while tending my collection of orchids.
What is the probability that I will start to repot my orchids in rockwool next year?
Well, I don't especially want to put a number to it, but I'd certainly say "low".
Now, that may be my belief. Suppose I now ask all my friends who also know
about probability and rockwool what value they would give to:
p(Paul repots his orchids in rockwool next year)
I honestly think they too would give an answer "low". And if we all gave numbers,
these would cluster around the low end of the 0-1 scale.
This suggests that there is a consistent empirical relation system in action even
with this "subjective" value. Similarly to the behaviour of people on judging height
I gave in my earlier example, this suggests that maybe some consistency
can be achieved in using probability as a measure on a property of the world
(perhaps a "measure of propensity").
The consistency arises because this is a fairly well understood situation (amongst
people who know me, rockwool and probability). So I still claim (provisionally)
that (for me) it is helpful to think of probability as a measure. It is an "indirect"
measure, and as such is never (?) possible to measure accurately.
You obtain a probability, then, by:
Preferably measuring and understanding the situation you are interested in (count the
balls in the bag, check they are all the same mass and size, check the surface
texture,....);
or
Repeating the situation you are interested in (repeatedly sampling balls from the bag),
and you can work out how many samples you need for a given level of accuracy);
or
Ask an expert _provided_ the question is cognitively meaningful and within
that expert's domain of expertise. Better still ask several experts.
Now, as I understand it, the Big response from a Bayesian may be to question
the need for consistency in the last. Then the sand begins to fall away around my
feet. _Not_ because I don't think my argument holds up, but because if
several people can have significantly different probabilities for the same
situation, then I just do not know who to believe. I do not know how to
trust the figures I get.
Best regards
Paul
-- ====================================================== Paul J. Krause Philips Research Laboratories Crossoak Lane Redhill, Surrey RH1 5HA United Kingdom Tel: +44 (0)1293 815298 Fax: +44 (0)1293 815500 mailto:krause@prl.research.philips.com =======================================================