Self-deterministic probability distribution

[Loren Booda]

Do linear probability distributions in pure mathematics ever evolve chaotically? In other words, can random statistics tend to reinforce certain singular values nonrandomly? I am reminded of quantum models that favor definite, discrete outcomes.

 

[Enuma_Elish]

I'm not familiar with chaos theory, so I may be committing a type III statistical error (a.k.a. "answering the wrong question").

Could you get there by defining the distribution in the k^th iteration as a function of the outcome(s) in the k-1^st iteration?

E.g., if I were to define the family of cumulative uniform distributions U_k(x) = x/a_k where a_k is the realization in the k-1^st stage (and say, a₀ = 1), then U_k --> 0 (at least in expectation).

Then, one can define a set of such families, each family having its own rule of path-dependency. (E.g., I can also define another uniform family V_k that converges to 1, and define the set as {U_k(U), V_k(V)}.) Finally, one can come up with a random selection rule over the set, so that the observed outcomes seem to randomly oscillate toward 0 and 1.

EnumaElish
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I would definitely have logged in as EnumaElish had PF administration awarded that account the privilege of posting replies, after I reset my e-mail address Tuesday, October 28, 2008.

 

[Loren Booda]

Let me express my concern as clearly as I can: what is the cardinality of all probabilities?

Sorry for the diversion on the way here, Enuma Elish.

 

[Enuma_Elish]

This blog contains a few related ideas:

http://quomodocumque.wordpress.com/2008/11/26/tom-leinster-on-entropy-diversity-and-cardinality/

EnumaElish

 

[Loren Booda]

E.E.,

Your response is about as right on as I could ask. The initial blog (as I think I understand its first half) and its references are intriguing and interesting. My sense tells me that fractals might be directly involved there.

The definition "cardinality as the ‘effective number of points’" is a good starting place. (I guess a "metric space" is one that can be assigned probability.) Exponential entropy, a new concept to me, compares well to cardinality. This seems the meditation between cardinality and probability I seek.