Talk:Information geometry

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[edit] Substance

"Intuitively, this says the distance between two points on a statistical differential manifold is the amount of information between them, i.e. the informational difference between them."

"Thus, if a point in information space represents the state of a system, then the trajectory of that point will, on average, be a random walk through information space, i.e. will diffuse according to Brownian motion."

"With this in mind, the information space can be thought of as a fitness landscape, a trajectory through this space being an "evolution". The Brownian motion of evolution trajectories thus represents the no free lunch phenomenon discussed by Stuart Kauffman"

These lines are meaningless to me. Can we replace them with something that carries meaning? The first one is indeed not intuitive; the second one introduces a system that I don't remember being described, following a path which I don't understand, and asserts an implication which is not at all obvious; and the third one goes even farther afield by making a connection to intelligent design of all things. Is it supposed to be a joke? A5 22:07, 26 April 2007 (UTC)

Was that supposed to be a joke? Making a connection to intelligent design?!?!

First, 1. if you integrate, from one point to another (x0 to x1), so as to find the length of that path, over a geodesic, what you get is an expression that includes -ln(p); information, and that value represents the difference between probability models x0 and x1. (points on the space represent different probability models)

Second para: This is intuitively obvious when you understand what the geometry represents, by understanding the first paragraph. If the state of a system changes at an average rate of x bits of information, and the metric for distance represents bits of information, as stated in the first paragraph, then all points on a circle of radius dx (i.e. infinitesimal radius) centered on the state of the system at time t are equally likely to be the state of the system at time t+dt. This is equivalent to saying that, on average, trajectories will follow a random walk.

Stuart Kauffman proposed the "no free lunch theory". where the geometry of the space; the particular metric that defines the geometry; represents the "fitness landscape". In information geometry, a unit evolution on the landscape - i.e. one bit of information change in the state of the system, would amount to one unit of distance along the manifold. Given furthermore, as is postulated by the no free lunch theorem, that the direction of evolution (angle of travel through the space) at any given moment, is, at best, random, to represent the no free lunch theorem on this manifold would be to, at each time differential dt, travel (on average) a distance differential dx, in a random direction. I.e., a random walk.

Kevin Baas^talk 16:11, 29 April 2007 (UTC)