The n-Category Café

Does the concept ‘concurrent information processing’ provide an explanation for the similarity between learning with nonparametric statistical models and physical field theories?

In Bayesian Field Theory: From likelihood fields to hyperfields, Lemm sets up the situation as follows:

For empirical learning it is convenient to distinguish three groups of variables:

• 1. observable (visible) independent variables $x$ representing ‘inputs’ or ‘controlled causes’,
• 2. observable (visible) dependent variables $y$ representing ‘outputs’ or ‘measured effects’,
• 3. not directly observable (hidden, latent) variables $\varphi$ describing the possible (pure) ‘states of Nature’ considered in the model under study.

Now the posterior predictive density is

$p(y\mid x,D,D_0)=\int d\varphi p(y\mid x,\varphi )p(\varphi \mid D,D_0)$, where $D$ is the training data and $D_0$ is the prior knowledge.

Lemm continues:

In nonparametric models where $\varphi$ is a function the integral over $\varphi$ stands for a functional integral. Such integrals are mathematically well defined for Gaussian processes (‘generalized free Euclidean fields’ or ‘generalized Brownian motion’ in the language of physics), but for more general processes (‘(self–)interacting fields’) a general mathematical definition of functional integrals is still an unsolved problem. One possible approach is to define a non–Gaussian functional integral by renormalization procedures in perturbation theory or on a lattice, another simpler possibility is to discretize the function $\varphi$ in some appropriate basis so the functional integral becomes a standard (but typically still extremely highdimensional) integral.The remaining highdimensional integrals can either be approximated by Monte Carlo methods or by the method of steepest descents (Laplace approximation), in this context known as Maximum A Posteriori (MAP) approximation…In principle one can also go beyond the MAP approximation by using the MAP solution as starting point for a perturbation series, for example, graphically expressed by Feynman diagrams.

Does all of this make sense from the physics side?

Dear David Murphy,

I’m interested in reading you paper with Robin Knight. I would very much appreciate, if possible, to receive a copy of it.

christinedantas $#at#$ yahoo.com

Thank you,
Christine

I can confirm that this connection was apparently well enough known for my lecturers to discuss in my undergraduate classes (in ‘94).

To illustrate the issue they’d start by describing some system (eg a vending machine) using a formalism like the pi-calculus, then “break” it by asking us to consider a new situation with some extra physical parameter (eg time) which wasn’t amenable to the model.

After considering a more few examples (realistic bounds on the in-degree of processes, heat dissipation, hardware design, …) they made the case that the logical conclusion was to consider physics as a “model” of concurrency — but that for this line of thought to be useful to computer scientists, we’d prefer a concurrent model of physics! (as Christina suggests!)

What strikes me about Abramsky and Coeke’s papers is the suggestive and amazingly simple new formalism. You can really imagine specifying systems in it (well, at least after the nice clear examples they give), which is as far as I know a first for this area.

So it seems to me (although I have never done this for a living) that this exciting area is still in it’s infancy. Eg one of the operational semantics for the pi-calculus was a lot of fun – the “chemical abstract machine” (CHAM) was based on the notion of processes as a reactive gas which would perform computation as it was “heated” and “cooled”. So what about a “physical abstract machine” — would I be right in guessing that this is close to the ideas you were asking about Christina? (maybe at a higher mathematical level than I’ve described?)

I just happened on this post by David Murphy (hello David - long time no see).

If only David M had looked at my short discussion paper which David Corfield originally referenced, starting this thread, he would have seen that much of it was explicitly about Petri’s pioneering contributions in this direction!

This little piece attracted the attention of the Petri net community, who have invited me to speak at their annual conference in July.

The current developments are in my view very exciting, and going much further in both scope and depth than those pioneering contributions did.

For example, there will be a session on these links between Computer Science and Physics at the MFPS conference next month. See:

http://www.math.tulane.edu/~mfps/program23.html

See also:

http://se10.comlab.ox.ac.uk:8080/FOCS/CKCinOXFORD_en.html

And the QICS project, a European project which has just started up, involving both Computer Scientists and Physicists. See:

http://web.comlab.ox.ac.uk/oucl/work/ross.duncan/qics/

I guess what I was trying to say in my original post was that Carl Adam Petri thought he was doing that already… Perhaps we have better tools now

According to a paper by Sokolowski, “Directed topology and concurrency - a short overview”, local po-spaces generalize several classical models of concurrency. In increasing generality:

Petri nets -> transition systems -> higher dimensional automata -> cubical complexes -> local po-spaces…

Best,
Christine