### More on information geometry

Some blogs use different categories to sort out their posts. However, in my experience, things that interest me enough tend to show themselves to be related somewhere down the line. A while ago I was pondering the question -

How much of the mathematics used in physics is describing our knowledge and ways of observing and intervening, and how much the physical world itself?- in the context of Caves and Fuchs' interpretation of quantum theory. Now I see Ariel Caticha has an article trying to understand general relativity in terms of information geometry:

The point of view that has been prevalent among scientists is that the laws of physics mirror the laws of nature. The reflection might be imperfect, a mere approximation to the real thing, but it is a reflection nonetheless. The connection between physics and nature could, however, be less direct. The laws of physics could be mere rules for processing information about nature. If this second point of view turns out to be correct one would expect many aspects of physics to mirror the structure of theories of inference. Indeed, it should be possible to derive the “laws of physics” appropriate to a certain problem by applying standard rules of inference to the information that happens to be relevant to the problem at hand.Elsewhere, work is underway to generalise information geometry to the infinite dimensional spaces used in nonparametric statistics. There are important papers at Jun Zhang's site, including 'Nonparametric information geometry: Referential duality and representational duality on statistical manifolds.' Zhang is a psychologist who uses this mathematics to model psychometric testing.

## 3 Comments:

Another important source for information geometry is the group of shun-ichi amari at the RIKEN institute for neuroscience. http://www.brain.riken.jp/labs/mns/amari/home-E.html

Shalizi's notebook also has some information on this research group. They developed information geometry in order to aply it to neuronal networks but I believe Amari and Nagaoka's 'methods of information geometry' is one of the important general books in the field.

Our standard statistical decision procedures are compromises, effecting a trade-off between competing desirable objectives. (eg, we want to minimize both the chance of rejecting true hypotheses and the chance of accepting false ones, and these two chances are typically inversely-related). Different societies or cultures, valuing the consequences of inference errors differently to ourselves, could well decide these trade-offs, and hence the corresponding statistical decision procedures, differently.

It would therefore strike me as very surprising indeed if our particular compromise procedure turned out to reflect some underlying physical reality.

-- Peter

It would therefore strike me as very surprising indeed if our particular compromise procedure turned out to reflect some underlying physical reality.If this line of reasoning were to work at all, it would have to concern an ideal knower. If I understand the Cave-Fuchs position, much of the machinery of QM is about the manipulation of maximal states of information, the weirdness of QM largely resting in the fact that these maximal states are not complete.

Outside QM, I'm intrigued to see similar constructions occurring in a psychologist discussing Referential Duality and Representational Duality in the Scaling

of Multi-Dimensional and Infinite-Dimensional Stimulus Space, Caticha, a Bayesian, discussing space-time, and a physicist talking about Counting probability distributions: Differential

geometry and model selection with two psychologists.

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