"Edward Green" <spamspamspam3@netzero.com> wrote in message
news:1150175040.701621.269320@i40g2000cwc.googlegroups.com...

Nobody asked me, but...

I've been thinking about what exactly the so-called Bayesian point of
view comprises, and how it differs from the so-called frequentist view.
First, some ways they may be alleged to differ but don't:

(1) Bayes used a particular equation to update:

The Bayesian trope alludes to a fundamental equation, which gives our
a-postiori probabilities based on prior probabilities, and an
observation. This equation is a standard application of conditional
probability and contains no features unique to Bayes.

(2) Bayes considers unique, non-repeatable events:

This may be more controversial, but I allege there is no real
distinction here. Any real event, a particular roll of a particular
die, is a non-repeatable event. I can only roll a particular die at a
particular time once. As an idealization I may claim that I can repeat
this experiment many times with the same die assumed to obey the same
distribution, or with a population of identical dies at the same time
assumed to obey the same distribution. But I can ideally do this with
any unique event: I can abstractly postulate an ensemble of universes
conforming to some distribution of initial conditions, even if I only
can observe one instance.

What then is different? I think it is merely:

(3) The Bayesian is willing to guess.

Confronted with a unique event, which we ideally want to consider a
single instance of an ensemble of identically distributed events, the
Bayesian is willing to assign probabilities based upon his sum of prior
knowledge in an otherwise undefinable way, e.g. "given my experience
with this kind of person, I'd say there is about a 70% chance he is
lying". The alternative approach would be to refuse to make any
estimates in the lack of almost certain knowledge. This may be
satisfying for the purist, but is the opposite of pragmatic: we are
often required to act without the luxury of nearly perfect knowledge --
including perfect quantification of our ignorance.

The question is whether we are willing to accept uncertainty about
uncertainty.

Uncertainty about uncertainty must be taken in account of course. My
question is rather the opposite: what is the true meaning of frequentism.
Frequentism doesn't tell what a physics experimenter really wants to know -
a full probability estimation on which a confidence interval may be based.
An interesting discussion on this subject can be found in
http://arxiv.org/abs/hep-ex/0002055

Cheers,
Harald

From Osher Doctorow

Edward Green typed:

What then is different? I think it is merely:
(3) The Bayesian is willing to guess

The Bayesians are willing to guess, but usually only within very
restricted contexts. You mentioned conditional probability, the
"probability of Y given X =3D x" for example, where X, Y are random
variables and x is a real value of X. At the graduate level of
mathematical probability, it turns out that conditional probability and
conditional expectation reduce to or equal the Lebesgue-Radon-Nikodym
or Radon Derivative, but at the intermediate and elementary levels the
practical translation of this is more obvious and simpler, namely that
things are based on expressions of the form:

1) y/x, where y =3D P(AB) or f(x, z), x =3D P(A) or fX(x)

where P( ) is probability, fX(x) is the (marginal) probability
density function (pdf) of random variable X at x, f(x, z) is the joint
pdf of X and Y at (x, z).

My wife Marleen and I have tried since 1980 to convince Bayesians and
other users of conditional probability to consider study of the
expression:

2) 1 + y - x, y =3D P(AB) or f(x, z), x =3D P(A) or fX(x)

which is Probable Influence/Causation (PI). There are two contexts in
which 1 + y - x arises. First, if one replaces division in y/x by
subtraction as in 1 + y - x, then the difficult case of x =3D 0 and the
blow-up of y/x very close to x =3D 0 (from the right) disappears.
Secondly, if we define the set (A-->B) =3D (AB' )' =3D A' U B, in perfect
analogy with logical a-->b =3D ~(a ^ ~b) =3D ~a V b where AB' is the
intersection of sets A and B' and U is set union ("and/or") and ~ is
negation (not) and a, b are propositions and ^ is conjunction ("and")
and V is disjunction (and/or), then P(A-->B) =3D 1 + y - x with y =3D P(A=

and x =3D P(A).

The Bayesians in the University of California (several branches) and
Stanford whom we contacted (quite a few of them) refused to have
anything to do with 1 + y - x on the incredible grounds that "if y/x
ain't spoiled, then don't fix it" more or less. Bayesians in
Artificial Intelligence around the country had similar reactions or
simply didn't reply, and ditto with top reliability engineers.

The kind of mental ossified bureaucratic thinking of a person who
refuses to study 1 + y -x because it isn't the same as y/x is so
primitive that I would be tempted to describe it as apelike. Since
these people aren't technically apes but are members of university
Academia, I describe them as Ingenious Imitators since they haven't
even gotten to the level of slightly varying an axiom or definition or
postulate, unlike Non-Euclidean geometers (though the latter have their
own difficulties). A Creative Genius should at least have minimal
interest in changing axioms, definitions, postulates.

If you look at my previous postings and threads in sci.physics, and in
sci.stat.math and geometry.research and math-history-list of Math Forum
and some papers that did manage to get published, you will hopefully
see that 1 + y - x does indeed lead to quite different results and
predictions and theories from y/x.

Osher Doctorow