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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.
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509:. Unfortunately, that revision removed a bunch of simply-worded, easy to understand text, replacing it with the current impenetrable raft of goo. So part of the effort must be to revive some part of the older content, and provide it as the intro. And certain parts of this article, e.g. the alpha connection, probably should be (??) moved to its own stand-alone article.
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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
460:
I plan to cut out large parts of this article; most/all of it recapitulates large parts of well-known differential geometry concepts that are well-explained in other WP articles. After cutting these out, I'll see whats left, and try to turn that into a real article. I figure someone will scream in
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I had to look at the history of this talk page to see what you were talking about. Yeah, it was strange nonsense of some kind. However, in general, one should not ever delete stuff from the talk pages (unless moving it to an archive page). It would have sufficed to just put a section heading on it,
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I removed most of the material and started on a rewrite to make this more readable. It still needs a lot of work, though. I couldn't get that one paragraph to format correctly, which is due to my inexperience with
Knowledge. I'm a geometer by background, so hopefully someone with more knowledge of
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I was interested in information geometry, read part of the Amari book and wanted to share the basics. I'm not an expert, though. You get the drift of what you're saying. You can probably do a better job than me. So please go ahead. Generally I think for the sake of the flow of thoughts to create
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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
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I agree with your assessment. A number of the concepts of differential geometry are summarized here so briefly by intransparent formulas that it looks more like note-cards than an encyclopedia article. It would be better to replace them by more generally worded descriptions with pointers to more
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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,
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Ugh. That still leaves behind a huge raft of formulas with no intuitive explanation given at all. Even the very first section gives a standard information-theoretic argument that can be found in textbooks, but here it is so awkwardly worded that I cannot figure out quite how to fix it. The next
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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
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I am slowly working on this, but welcome help. If no one is opposed, I will continue to remove whole sections as I re-write since the current later sections are pretty randomly organized. I know
Information Geometry rather well, but this is my first wiki article, so I welcome comments and
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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.
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Any objections to removing all of the following? The comments are somewhat bizarre and unrelated to the true nature of information geometry. Furthermore, they are generally quite old and unsigned. Please make an effort to sign your posts to facilitate discussion.
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in-depth articles. Information geometry is a synthesis of statistics, information theory and differential geometry, but it spends all the time on the latter, trying to do a hurry-up, technical, but inadequate precis of a semester course in differential geometry.
361:"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"
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the right mindset it is better to repeat the essential and in addition add a reference to the general topic. This benefits most of the people dropping by because they want to learn and thus are rather new to the topic.
358:"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."
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This article is in urgent need of a total rewrite. As it exists, it is surely one of the worst in all
Knowledge. I will add it to my list, but my time is very limited. Can anyone else take a crack at it?
251:
A lot of the statements made in this article are not strictly true under all circumstances. Thus, many of the statements made should be modified to mention the scope of circumstances that the claims are
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Removed most of stuff on tangent spaces and metrics and tensors, as this is covered in any & all textbooks, and have nothing to do with info geom. Will try to attack rest of article later.
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355:"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."
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statistics can help with the statistical background and flesh out the applications a bit more. If need be, I can try to provide citations for where IG is used in those fields.
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These lines are meaningless to me. Can we replace them with something that carries meaning? The first one is indeed
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sections are hardly better. As I read through John Baez's posts, I'll try to clean this up, may take a while.
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I agree with these assessments, see also the remarks below under "Removing most of the material".
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supposed to be a joke? Making a connection to intelligent design?!?!
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Hi, would someone be so kind to add some (simple) examples?
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These are all in Amari's book, listed in the references.
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to separate if from the reasonable discussion below.
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