----- Original Message -----
From: "Alex Yu" <[EMAIL PROTECTED]>
To: "Shareef Siddeek" <[EMAIL PROTECTED]>
Cc: <[EMAIL PROTECTED]>
Sent: Thursday, March 01, 2001 5:01 PM
Subject: Re: probability definition
1. Very Interesting. Would it be possible to get a copy of your paper
on Probability. I gave a review of "The Meanings of 'Probability'"
several years ago for our ASA chapter, and would like to redo it.
> For a quick walk through of various prob. theories, you may consult
"The
> Cambridge Dictionary of Philosophy." pp.649-651.
>
> Basically, propensity theory is to deal with the problem that
frequentist
> prob. cannot be applied to a single case. Propensity theory defines
prob.
> as the disposition of a given kind of physical situation to yield an
> outcome of a given type.
>
> The following is extracted from one of my papers. It brielfy talks
about
> the history of classical theory, Reichenbach's frequentism and
Fisherian
> school:
>
> ------------
> Fisherian hypothesis testing is based upon relative frequency in
long
> run. Since a version of the frequentist view of probability was
developed
> by positivists Reichenbach (1938) and von Mises (1964), the two
schools
> of thoughts seem to share a common thread.
2. Von Mises (1957) quotes Johannes von Kries and goes on the address
the use as "I shall assume therefore a definite probability of the
death of Caius, Sempronius or Titus in the course of the next year".,
as support of his concept of "probability in a collective". He does
include the "single event" as part of his "collective". He then
states, "The term 'probability' will be reserved for the limiting
value of the relative frequency in a true collective which satisfies
the condition of randomness." With respect to Caius, Sempronius and
Titus" he was considering the collective of aged rulers of Rome.
>However, it is not necessarily
> true. Both Fisherian and positivist's frequency theory were proposed
as
> an opposition to the classical Laplacean theory of probability.
3. My reading of Fisher was that he opposed the Laplacian view because
it had no mathematical basis, Bayes however did, and was fully
accepted.
> In the
> Laplacean perspective, probability is deductive, theoretical, and
> subjective. To be specific, this probability is subjectively deduced
from
> theoretical principles and assumptions in the absence of objective
> verification with empirical data. Assume that every member of a set
has
> equal probability to occur (the principle of indifference),
probability
> is treated as a ratio between the desired event and all possible
events.
> This probability, derived from the fairness assumption, is made
before
> any events occur.
>
> Positivists such as Reichenbach and von Mises maintained that a very
> large number of empirical outcomes should be observed to form a
reference
> class. Probability is the ratio between the frequency of desired
outcome
> and the reference class. Indeed, the empirical probability hardly
concurs
> with the theoretical probability. For example, when a dice is
thrown, in
> theory the probability of the occurrence of number "one" should be
1/6.
> But even in a million simulations, the actual probability of the
> occurrence of "one" is not exactly one out of six times. It appears
that
> positivist's frequency theory is more valid than the classical one.
> However, the usefulness of this actual, finite, relative frequency
theory
> is limited for it is difficult to tell how large the reference class
is
> considered large enough.
4. The idea of a "limiting condition" is based on the same
understanding of differential calculus and infinite series. That is a
limit is reached, not on the value of N, only that some value
converges to a limit as N increases. This does not depend on any
arbitrary large value of N
5. von Mises also based his position on the laws of probability, which
can be defined as a "natural" outcome from the frequentist view, where
limiting occurs as a converging value of a ratio. He differentiated
between the "actual occurances" and as a "thought experiment". It was
the latter that he was using.
> Fisher (1930) criticized that Laplace's theory is subjective and
> incompatible with the inductive nature of science. However, unlike
the
> positivists' empirical based theory, Fisher's is a hypothetical
infinite
> relative frequency theory. In the Fisherian school, various
theoretical
> sampling distributions are constructed as references for comparing
the
> observed.
6. My reading of the historical record is that this was the K. Pearson
school that did this. Fisher stuck to the Uniform, Normal and Poisson.
> Since Fisher did not mention Reichenbach or von Mises, it is
> reasonable to believe that Fisher developed his frequency theory
> independently.
7. Fisher was aware of many who challenged his views, but chose not to
respond, except to K and E Pearson. Anyone who challenged his views
like von Mises did on likelihood (back in 1930), was either to be
ignored or challenged by an exchange of letters on Fishers home
grounds. von Mises was not in his "backyard" and really did not offer
anything for Fisher to expand on. Fisher was a very private soul.
> Backed by a thorough historical research, Hacking (1990)
> asserted that "to identify frequency theories with the rise of
positivism
> (and thereby badmouth frequencies, since "positivism" has become
> distasteful) is to forget why frequentism arose when it did, namely
when
> there are a lot of known frequencies." (p.452) In a similar vein,
Jones
> (1999) maintained that "while a positivist may have to be a
frequentist,
> a frequentist does not have to be a positivist."
8. Since you include positivist and frequentist as being the same
above, what is the difference that Hacking is refering to?
>
DAHeiser
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