Exotic Probabilities
odd74.proboards.comIt 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)
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.
> 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?
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.
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.
> 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].
Sum to 0 if probabilities are nonnegative scalars seem to be cumbersome. If probabilities are mathematical entities that are free to choose, the sum to 0 "can be played" by quite a few abelian groups.
I think it's especially important to be able to make a distinction between truth and falsehood however. If they are both 0, this is difficult. :-)
black-scholes aka the financial equation that uses normal distributions to model non-gaussian systems.