
Exotic Probabilities - ergoproxy
http://odd74.proboards.com/thread/11196/prob-stat-101?page=1&scrollTo=169139
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pash
It is intuitively obvious that requiring probabilities to sum to 1 is rather
arbitrary (any other number would seem to do as well), and this is confirmed
by many of the alternatives to Kolmogorov's axiomatization of probability
theory.

For example, one may formulate axioms of probability theory as an extension of
ordinary two-valued logic. From a handful of elementary axioms capturing what
it means for degrees of belief in combinations of propositions (on some
evidence) to be consistent, you can derive a differential equation whose
solution is a functional representation of what is often taken as a definition
of conditional probability in mathematical probability theory:

    
    
        c * f(X and Y | Z ) = f(X | Y and Z) * f(Y | Z)
    

Here _c_ is the constant of integration in the solution to the differential
equation, and it turns out also to be the sum of the probabilities of all
mutually exclusive events. It is not determined by any side conditions, so it
is set to 1 by convention; but any other value would do. [0]

Negative probabilities are different beast, best viewed as algebraic
extensions to probability theory in the manner that, e.g., the integers are
algebraic extensions of the natural numbers (i.e., by including additive
inverses). But I am not very knowledgeable on the subject, so I will say no
more.

0\. For details, see Cox (1961), /The Algebra of Probable Inference/ (recently
back in print) or Jaynes (2003), /Probability Theory: The Logic of Science/.
Both are excellent books, the former covering probability theory and entropy
as extensions of logical reasoning, and the latter covering all that and much
else of Bayesian probability and statistics.

~~~
thaumasiotes
> It is intuitively obvious that requiring probabilities to sum to 1 is rather
> arbitrary (any other number would seem to do as well)

Well, you have basically two choices -- one and zero. Any nonzero real number
is trivially equivalent to 1.

It's not obvious to me, though, that 0 would work just as well?

~~~
TeMPOraL
Not only 0 and 1 are Schelling points - default options you can assume pretty
much everyone would chose - they're also special in the way that they define a
(positive) range where numbers always stay inside that range under
multiplication. Two numbers between 0 and 1 multiplied together will always
give a number that's also between 0 and 1. That's why a lot of places in math
like to transform domains into the 0...1 range.

~~~
closed
This. Another way to express it is that for an interval [a, b], multiplying
two numbers c, d within that interval will be within [aa, bb], and for 3
numbers [aaa, bbb], but it's super convenient to that for a=0 and b=1 these
are always [0,1]. Other numbers would work, but would become cumbersome.

~~~
thaumasiotes
> for an interval [a, b], multiplying two numbers c, d within that interval
> will be within [aa, bb]

This only works when a is nonnegative. For example, the range of [-1, 1] under
self-multiplication is [-1, 1], not [1, 1].

------
DINKDINK
black-scholes aka the financial equation that uses normal distributions to
model non-gaussian systems.

