>Throughout the 20thcentury, probability has come to mean to two very
>different things:
>
>1) Frequency (a.k.a. a physical probability)
>2) Uncertainty (a.k.a. a subjective probability)
>
>It sometimes amazes me at how often this is pointed out, agreed upon, and
>then forgotten.
I don't agree with this at all.
Probability IS neither of these things.
Probability is a mathematical theory.
The mathematics of probability can be USED TO MODEL frequencies. It can be
used to model degrees of belief.
You can use that definition if you like, and I will PROBABLY understand
what you mean. But, I still assert that what is commonly referred to as
probability is first and foremost frequency theory which may also, on some
occasions, be reasonably used to represent a degree of belief
The nice thing is that if you have repeated observations on a phenomenon
which exhibits stable frequencies, and if you have a nondogmatic prior
distribution for your uncertainty about the phenomenon, then your posterior
distribution will be very similar to the empirical frequencies.
A variant of this result has been formalized as de Finetti's theorem.
What nondogmatic means is:
- it is not too unlikely that the events are what Bayesians call
exchangeable and the frequentists call independent and identically
distributed with an unknown frequency distribution;
- your prior distribution does not put nearly all of its weight somewhere
other than where the actual frequencies lie.
Of course, the higher the dimensionality of the distribution you are trying
to estimate, the more observations it will take for the posterior
distribution and the frequencies to become close to each other.
Perhaps the major problem with frequency theory = probability theory is
that, at least by mu calculations, not all probability can be reduced to
frequency. In particular, how would you go about calculating the
probability that a prior is nondogmatic? Inductive reasoning is not truth
functional or mathematical, but it very definitely has something to do with
probability
>The problem with the phrase "random variable" is that it seems to rather
>deliberately run the two concepts together (i.e. random = uncertain and
>variable = frequency).
Not exactly.
Frequentists don't run the concepts together because they won't apply
probability to problems characterized by what you call uncertainty. They
are very clear about this.
What they say and what they do are two different things. Frequency theory
is called probability theory because it was first devised as a technique
for calculating odds for single events. Frequentists also continue to use
words like 'likelihood' and 'confidence', which is pretty misleading if you
are not talking about uncertainty. I also find it really hard to think of
a Fisherian significance test as anything other than a method for managing
uncertainty.
Strict subjectivists deliberately run the concepts together because they
don't think there are two concepts. There is only one kind of "thing" to
which probabilities can be applied -- processes or events or outcomes about
which we are uncertain. For events exhibiting stable frequencies,
nondogmatic subjectivists will come to agree closely on predictions of
future events of the same kind. For events not exhibiting stable
frequencies (e.g., one-of-a-kind events), reasonable subjectivists can
agree to disagree. However, it is very difficult, and subjectivists say
futile, to try to make a clear distinction between "frequency" type
phenomena and "degree of belief" type phenomena.
It's not so hard to distinguish frequency and belief if you allow the
terminology be context dependent. If you are concerned with the series or
population, then it is frequency. If you are concerned with the single
event, it is uncertainty. BTW, a "degree of belief" is not a phenomenon.
So why bother trying?
It's nice to be understood.
No one is sweeping anything under the rug. There is just a philosophical
disagreement on the kinds of phenomena to which probability theory is
legitimately and usefully applied. However, people persist in labeling it
as an argument over what probability "really is," rather than an argument
over the class of pheonmena to which one person or another is willing to
apply probability theory. That's where we get into fights.
Well, I still think using ambiguous term such as 'probability", when either
"frequency" or "uncertianty" is meant is asking for trouble.