Re: Axioms of elementary probability

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear David,

On structure: Domain L (say) just needs to be distributive lattice - not
Boolean algebra.

The axiom P(top) = 1 looks an obvious dual to P(bottom) = 0, but there's
a lot to be gained from considering P with codomain [0,infinity] and
forgetting P(top) = 1.

Maps P: L -> [0,infinity] satisfying P(0) = 0 and the third (modular)
law are called valuations - I believe this dates back to Birkhoff's book
on lattice theory. In the case where L is a frame (complete lattice,
with binary meet distributing over all joins) and P is Scott continuous,
P is called a continuous valuation. These have been studied in domain
theory (Jones, Plotkin: probabilistic power domain) and general locales
(including by Heckmann, by Coquand and Spitters and by myself).

More generally, the domain of P can fruitfully be any commutative monoid
M. There is a universal valuation L -> M(L) in this generalized sense,
with M(L) got by adjoining finite monoid structure to L and forcing the
two laws.

Coquand and Spitter cite an interesting construction of M(L) by Horn and
Tarski. Let L* be the set of finite lists over L, and define a preorder
on L* by

  [x_i]_{1 in I} <= [y_j]_{j in J}

if for every natural number k,

  \/{x_K | K subseteq I, |K| = k} <= \/{y_K' | K' subseteq J, |K'| = k}

where x_K = /\{x_i | i in K} etc.

Then M(L) is isomorphic to L*/(equ reln corresponding to <=).

The relations holding in M(L) are what can be proved from the theory.
You give a ternary inclusion-and-exclusion for P(A u B u C). If you
bring all the negative terms from right to left, it will still hold in
M(L), and can be generalized from ternary to n-ary. I think you will get
the dual (for P(A n B n C)) by considering L^op.

Another interesting relation, which can be used in proving the
Horn-Tarski result, is this:

   Sigma_{i = 0}^{n-1} x_i
      = Sigma_{k = 1}^{m} \/{x_I | I subseteq {0, ..., n-1}, |I| = k}

Regards,

Steve Vickers.

References:

Jones & Plotkin: "A probabilistic powerdomain of evaluations", LICS'89.
Horn & Tarski: "Measures in Boolean algebras", Trans. Amer. Math. Soc.
64 (1948)
Heckmann: "Probabilistic powerdomain, information systems and locales",
MFPS VIII, Springer LNCS 802 (1994)
Vickers: "A localic theory of lower and upper integrals", Math. Logic
Quarterly 54 (2008)
Coquand & Spitters: "Integrals and valuations", Journal of Logic and
Analysis 1:3 (2009

David Espinosa wrote:
>
> Here's a question about elementary (naive, finitist) probability.
> The proper, self-dual axioms for elementary probability are presumably
>
>  P(0) = 0
>  P(X) = 1
>  P(A u B) + P(A n B) = P(A) + P(B)
>
> P's domain is a boolean algebra.  P's codomain is [0,1].
> What kind of algebraic structure is [0,1] in this case?
>
> What can we prove from this theory?  The best I can think of is inclusion /
> exclusion:
>
>  P(A u B u C) = P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C) + P(A n
> B n C)
>  P(A n B n C) = P(A) + P(B) + P(C) - P(A u B) - P(A u C) - P(B u C) + P(A u
> B u C)
>
> Thanks,
>
> David
>
>
>
>