I was afraid a frequentist would respond with something like this. So here
is the nickel answer.
There are two problems with claiming Bayesian probabilities are physical:
1) In a Bayesian application we manipulate probabilities. If we claim they
are physical (relative frequencies), then we need to include our confidence
in them and compute our confidence in the inferred probabilities (it is not
same). I wrote several papers on how to do this in Bayesian nets in the
early 90's and never came up with a totally satisfactory method. Note that
Fisherian applications have totally do to with confidence in relative
frequencies.
If we just accept the fact that the probabilities are coherent beliefs
(often obtained at least partially from data), we have no need for this
confidence. The inferred probabilities are our beliefs; we need say no more.
2) Even if (1) is overcome, we now have a problem when the probability is
clearly not physical. What about a decision theory application where I need
P(this stock will go up tomorrow). The stock is not `typical' of any
population. You could work hard at fitting it into your relative frequency
mindset by imagining possible worlds, etc. But to what end are you taking
on all this baggage? As far as I am concerned, frequentists cling
tenaciously to populations because they grew up with probability courses
dominated by the von Mises' theory and the Kolmogorov axioms. So everything
has to be related to some imagined population. But this simply is the not
way we assess many probabilities. Consider this: An older guy at my health
club told me he had acquired a recently popular drug. Every time I saw him
I asked him if he used it yet. He got tired of me bugging him, finally said
yes, and told me a preposterous story. I think the probability he really
used it is about .1. It is simply my belief. I would bet according to it,
but I never thought of an imagined population or possible worlds to
determine it. I would only to do this if a frequentist told me I have to in
order to make it a probability. This is why I say I am a `born-again'
Bayesian. I feel one must purge themselves of all this imagined population
talk to think clearly about probability.
I really do not want to discuss physical verses subjective probabilities
anymore. I have had that discussion 1000 times. However, I would appreciate
an opinion on my two choices.
Sincerely,
Rich
.At 09:09 AM 6/18/98 +0100, Paul Krause wrote:
>Rich,
>But when Joe comes thru the door he is just a sample point from all the
>other Joes in the world. The probabilities you put in as parameters in
>a network aren't for a specific Joe, but are for a "typical Joe".
>
>The difficulty I have with all this is that in the context of learning, your
>"Bayesian probabilities" gradually turn into physical probabilities. So in
order
>to validate and refine a probabilistic network, you start to rely less and
less on
>probabilities as beliefs and more and more as probabilities as attributes or
>properties of the real world. I fundamentally do not like the distinction
>between "Bayesian probability" and (call it) statistical probability - if you
>have an adaptive probabilistic network (in the sense of aHugin) when do your
>Bayesian probabilities suddenly turn into physical probabilities?
>So I still start from the premise that the "Bayesian probabilities" are a
matter
>of expert judgement that you may use as a starting point. Better still,
take a
>mean from two or three experts. Best, assess the value from real world data.
>
>An analogy. How do you assess the height of a person?
>
>1) Show an expert on people a photograph of the person and ask them how tall
>that person is.
>
>Ans. Not very reliable in general (I'm assuming there is no yardstick in
the photo).
>
>2) Show 100 experts on people a photograph of the person and ask them how
tall
>that person is. Take a mean.
>
>Ans. Actually a pretty accurate measure of the height of the person (this
has been
>demonstrated).
>
>3) Take a standard measure and go up the the person in real life and
measure their
>height.
>
>Ans. The most accurate (by definition, because we all agree on the standard)
>measure - although you need to specify whether it is their "morning"
height or their
>"evening" height that you are measuring.
>
>Analogously, I start with the premise that probability is a measure and
you get the
>value as best you can. Expert judgement is quite good - _provided_ the
expert is
>will calibrated. But their are better ways.
>
>So I make no distinction: probability is probability is probability.
>I think this is the route cause of why I (personally) don't wish to see
additional notation
>brought in.
>
>A perfectly valid response to this is: "you're not a Bayesian, so your
comments don't
>count". Which I'm happy to accept. So my prior on
>"Paul sending out a message on random variable is a mistake"
>was quite high.
>
>The posterior is even higher. So this is my final comment and I duck out now
>from being anything but a passive observer in this discussion.
>
>All best
> Paul
>
>neo wrote:
>
>> Dear Paul,
>>
>> Again, what you describe are classical statistical applications where we
>> have a population of individuals (sample space) with the uniform
>> distribution over them, and properties of these individuals are random
>> variables on this space. Other than the name itself, I do not believe
>> anyone has a problem with random variables in this context.
>>
>> In Bayesian applications, we are talking about P(Joe has Cancer). There is
>> no underlying population. So to make Cancer a random variable we have to
>> take the Cartesian product of the sets of values of all `variables' and
>> call this the sample space. Then `Cancer' can be considered a function on
>> this sample space, and we can write P(Cancer=yes). But are we not forcing
>> ourselves to fit the problem into some existing representation that does
>> not really suit it? What was wrong with just saying P(Joe has Cancer)?
>>
>> Sincerely,
>>
>> Rich
>>
>> At 11:10 AM 6/17/98 +0100, Paul Krause wrote:
>> >Dear All,
>> >
>
>
>
>--
>======================================================
>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
>=======================================================
>
>
>
>