Axioms of elementary probability

Axioms of elementary probability

[David Espinosa]

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

Re: Axioms of elementary probability

[Steve Vickers]

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
>
>
>
>

Re: Axioms of elementary probability

[Greg Meredith]

David,

Here <http://arxiv.org/abs/math-ph/0508006>'s an arXiv reference for the
"cottage industry" i was referring to.

Best wishes,

--greg

On Wed, May 13, 2009 at 12:59 PM, Greg Meredith <
lgreg.meredith@biosimilarity.com> wrote:

> David,
>
> To my mind there are three presentations of a "theory" of probability. Two
> arrive at essentially the same theory by somewhat different means; these are
> frequentist and Bayesian presentations of "standard" probability theory. The
> third comes from a completely different direction: quantum mechanics. i
> remember when i first encountered the Dirac presentation of QM and the
> interpretation of <a| M |b> as a probability amplitude. My first thought was
> -- hang on, doesn't that come with an obligation to prove that this aligns
> with (satisfies the axioms of) a theory of probability. In attempting to
> work that out for myself, i realized that it didn't; discovered a whole
> cottage industry of people who had made a similar observation; and argued to
> myself that of the various notions of probability put forward, this one
> enjoyed being rigourously employed in physical calculations verified to many
> decimal places.
>
> Best wishes,
>

Re: Axioms of elementary probability

[Greg Meredith]

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Re: Axioms of elementary probability

[RFC Walters]

Readers of the list may be interested in the following paper:
L. de Francesco Albasini, N. Sabadini, R.F.C. Walters, The compositional
construction of Markov processes, arXiv:0901.2434v1, 2009.

We believe that the identification of probability theory with measure
theory should be replaced with a theory based
on processes. To do this the theory of processes needs to be developed
categorically.
I have some comments on my web page about such a development.

RFC Walters

-- 
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/

Re: Axioms of elementary probability

[Jeff Egger]

When I took a graduate course in probability, my lecturer began with 
a rather fine speech about the relationship between probability and 
(finite) measure theory; in it, he discouraged identifying the two.  
His point was that, insofar as probabilistic phenomena occur in the 
real world, no mathematical theory can aspire to do more than model 
probability---and that, while (finite) measure theory has been very 
successful at modelling probability, it also has shortcomings.

Intrigued, I sought him out later for more thoughts on the subject.
In the ensuing conversation, I gathered two tidbits of information
which readers of the list may appreciate: that Gromov believes that 
the future of probability theory lies in bicategory theory; and that 
discontent with measure theory stems, at least in part, from its 
failure to adequately handle conditional probabilities.  

To be honest, the latter point heartened me even more than the first.
From a purely aesthetic point of view, it has always irked me that one 
can meaningfully assign probabilities to things which are not events;
I interpret this as meaning that the (standard) notion of event is too 
narrow.  Of course, it is also the case that the (standard) formula 
for a conditional probability may result in the indeterminate 0/0, so 
it would seem that [0,1] is also too small a codomain for the map 
"probability", even classically understood (i.e., not getting into the
"free probability" of Voiculescu). 

Cheers,
Jeff.

----- Original Message ----
> From: Ross Street <street@ics.mq.edu.au>
> To: David Espinosa <david@davidespinosa.net>; Categories <categories@mta.ca>
> Sent: Tuesday, May 12, 2009 2:53:13 AM
> Subject: Re: categories: Axioms of elementary probability
> 
> A couple of years ago, Voevodsky gave an interesting talk at the
> Australian Math Soc
> Annual Meeting (at RMIT. Melbourne) about a categorical approach to
> probability theory.
> Google told me about:
> 
>    http://www.math.miami.edu/anno/voevodsky.htm
> and
>    http://golem.ph.utexas.edu/category/2007/02/
> category_theoretic_probability_1.html
> 
> Ross
> 
> On 09/05/2009, at 4:02 PM, 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)

Re: Axioms of elementary probability

[Ross Street]

A couple of years ago, Voevodsky gave an interesting talk at the
Australian Math Soc
Annual Meeting (at RMIT. Melbourne) about a categorical approach to
probability theory.
Google told me about:

	http://www.math.miami.edu/anno/voevodsky.htm
and
	http://golem.ph.utexas.edu/category/2007/02/
category_theoretic_probability_1.html

Ross

On 09/05/2009, at 4:02 PM, 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)