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October 9, 2007

Progic IV

Posted by David Corfield

We’ve discussed matrix mechanics over rigs in many places over the years. I remember us toying with the idea that morphisms between rigs would allow us to pass in one direction or another between the corresponding mechanics. Perhaps this might give us some link between, say, the quantum mechanics supported by a space and its topology, the latter being all about path integrals with truth values.

For an easy example, if there’s a non-zero possibility of a particle propagating from A to B within a space then there must be a path from A to B within that space. The fun would really begin if we could reach higher homotopy. Can we couch the Bohm-Aharonov effect in these terms?

But if we wanted to get probabilities in on the act, we appear to be blocked by the fact that probability theory is not matrix mechanics over a rig. On the other hand, as John points out, at least in the case of finite probability spaces, we can invoke Durov’s generalized rings or algebraic monads. So why not look at morphisms between generalized rings?

Who knows what fun might be had passing along such morphisms, given that for the generalized ring known as the field with one element, 𝔽 1\mathbb{F}_1, it is claimed that

…a lot of statements in algebraic topology become statements about homological algebra over 𝔽 1\mathbb{F}_1.

What is homological algebra over the other generalized rings? And if

…the higher K-theory of 𝔽 1\mathbb{F}_1 must be the homotopy groups of spheres (p. 1),

what of the higher K-theory of other generalized rings?

Now a morphism α\alpha between monads, PP and QQ, gives rise to a functor, FF, between Kleisli categories. This functor is the identity on objects, and if ff is in Hom KleisliP(X,Y)Hom_{Kleisli P}(X, Y), so that it is a map between XX and P(Y)P(Y), then F(f)=α(Y)fF(f) = \alpha(Y) \cdot f, which is between XX and Q(Y)Q(Y), and so in Hom KleisliQ(X,Y)Hom_{Kleisli Q}(X, Y).

So we might want to look at morphisms between entries either of the right or left hand columns:

Monad Kleislicategory Identity Set +1 Partialfunction Powerset Rel Probability Conditionaldistributions Rmodule MatricesoverR\array{ \boldsymbol{Monad} & \boldsymbol {Kleisli category} \\ Identity & Set \\ +1 & Partial function \\ Powerset & Rel \\ Probability & Conditional distributions \\ R-module & Matrices over R}

But where does the geometry, e.g., Fisher information metric, get in on the act? And how is it passed between different Kleisli categories? Well, there is a distance (or better, family of divergences) for unnormalised densities (eqn. (2), page 5 of this), which passes to the ordinary one for normalised densities on restriction. And there is a Fisher metric in quantum information geometry. It would be worth seeing how this relates to the probabilistic case.

Changing tack, to give an example where the geometry may do some work in Progic, imagine we have a large data base containing incidence of disease. We have someone who smokes over 40 a day, but who is also vegetarian. Now we don’t have figures for heavy smoking vegetarians, but we do have them for vegetarians and for heavy smokers individually. Now we learn the probabilities for each class of person of the four conditions {±\pm heart disease &±\& \pm cancer}. So we have (estimates) for Pr(±H&±C|S)Pr(\pm H \& \pm C | S) and Pr(±H&±C|V)Pr(\pm H \& \pm C | V). The question is what should we say about Pr(±H&±C|S&V)Pr(\pm H \& \pm C | S \& V)?

The temptation is to draw a ‘straight line’ between the extreme points of the distributions for vegetarian and smoker. But what counts as straight? Here the idea of geodesics in the space of probability distributions appears. In these case, probability distributions over our two binary variables form a three dimensional surface, satisfying p(±H&±C)=1\sum p(\pm H \& \pm C) = 1. How would we join two points in it? There’s a good argument for taking logarithms of the coordinates, (logp(+H&+C),logp(+H&C),logp(H&+C),logp(H&C),(log p(+H \& +C), log p(+H \& -C), log p(-H \& +C), log p(-H \& -C), and looking for straight lines there.

Let me hint at one reason. If heart disease and lung cancer are independent conditional on being a vegetarian, and they are also independent conditional on being a heavy smoker, then you might think that intermediate distributions should continue possessing this independence property. But independence is expressed using a product, e.g Pr(+H&+C|V)=Pr(+H|V)Pr(+C|V)Pr(+H \& +C | V) = Pr(+H | V) \cdot Pr(+C | V), so we get

logPr(+H&+C|V)=logPr(+H|V)+logPr(+C|V),etc. log Pr(+H \& +C | V) = log Pr(+H | V) + log Pr(+C | V), etc.

Drawing a straight line between two independent distributions represented by log coordinates, gives us a family of independent distributions.

Posted at October 9, 2007 8:56 AM UTC

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3 Comments & 1 Trackback

Re: Progic IV

David Corfield wrote:

But if we wanted to get probabilities in on the act, we appear to be blocked by the fact that probability theory is not matrix mechanics over a rig.

That’s a provocative comment. Based on ideas from James Dolan, I’ve always espoused the opposite view. Yes, the rig [0,)[0,\infty) does not in itself incorporate the constraint that probabilities should sum to 1. But neither does the rig \mathbb{C} incorporate the constaint that amplitudes should have absolute values whose squares sum to 1! So, if probability theory isn’t matrix mechanics over [0,)[0,\infty), then why do you think quantum theory is matrix mechanics over \mathbb{C}?

Jim’s resolution to this puzzle is that numbers in [0,)[0,\infty) represent relative probabilities, just as numbers in \mathbb{C} represent relative amplitudes. We have to normalize these numbers to get actual probabilities. That’s what the partition function is for, in both statistical and quantum physics: it’s the normalizing factor.

I believe this is a consistent and sensible approach, though one would need to expand on what I’ve just said to really prove that.

It’s only much more recently, due to your progic project, that I noticed an alternate approach where we use one of Durov’s ‘generalized rings’ to incorporate — right from the start — the constraint that probabilities sum to 1.

I haven’t figured out how to do something similar in the quantum case.

(Here’s a puzzle for fans of dagger compact categories: when we think of quantum mechanics as matrix mechanics over \mathbb{C}, we first consider a dagger compact category where morphisms are \mathbb{C}-valued matrices. Then we note that unitary morphisms — those with

U U=1,U U=1U^{\dagger} U = 1, \qquad U^{\dagger} U = 1

are especially important, because they ‘preserve probability’. What happens when we work with the rig [0,)[0,\infty)? Do we get doubly stochastic matrices?)

Posted by: John Baez on October 9, 2007 4:48 PM | Permalink | Reply to this

Re: Progic IV

Now that’s close to the options I listed back here:

Thinking about kinds of morphism from XX to YY in {0,1}-valued matrices, we have:

  • a) Unnormalised: Rel;
  • b) Row normalised: Set;
  • c) Column normalised: Set opSet^{op};
  • d) Row and column normalised: Permutations.

Permutations, doubly stochastic matrices, and unitary transformations are avatars of the same idea.

So maybe an interesting question is whether there’s an interesting notion of row normalisation for complex matrices.

Posted by: David Corfield on October 9, 2007 6:33 PM | Permalink | Reply to this

Re: Progic IV

Is there any call in quantum mechanics for mm by nn matrices sending n\mathbb{C}^n to m\mathbb{C}^m such that vectors of amplitudes, the squares of whose absolute values sum to 1, are sent to similar vectors in m\mathbb{C}^m? These would have normalised rows.

Posted by: David Corfield on October 10, 2007 9:43 AM | Permalink | Reply to this
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Tracked: December 6, 2007 2:20 PM

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