Relationship between manifolds and random variables

  • #1
honestrosewater
Gold Member
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I am studying calculus and statistics currently, and a possible relationship between them just occurred to me. I was thinking about two things: (i) is a differentiable function from R to R a manifold, and (ii) in what way is a random event really unpredictable? So I don't know much about either one, but manifolds and random variables seem to be related by their local vs. global or short-term vs. long-term behavior. I don't know how closely complexity and randomness are related, but assume that they are closely related (or perhaps recast this in terms of information-content and description-length). It seems interesting to me that a manifold can be relatively complex/random globally/long-term but relatively simple/predictable locally/short-term, and the value of a random variable can be complex/random locally/short-term but simple/predictable globally/long-term. They seem to be duals or opposites. If you just look closely enough or long enough, they both get simple and predictable and more efficiently describable.

No? Comments? Is that a bad way to look at things?
 

Answers and Replies

  • #2
525
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manifolds and random variables seem to be related by their local vs. global or short-term vs. long-term behavior
It's possible you're mixing up manifolds with dynamical systems on manifolds, and random variables with stochastic processes. A more concrete link between manifolds (as topological spaces) and random variables is through integration rather than differentiation - using measure theory we can describe random events, discrete and continuous random variables and stochastic processes in a unified way.

A dynamical system can be viewed as a stochastic process where the state at a given time is conditional on certain unkown prior states - hence their behavior would be related. Does that help?
 

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