Timeline for Is there a combinatorial/topological treatment of statistical independence?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 12, 2017 at 21:33 | vote | accept | Chill2Macht | ||
| May 12, 2017 at 17:12 | comment | added | Henry.L | Read this article to see my comment about how hard it is to compute the $\Lambda$ construction for brownian motion. Yan, Catherine Huafei. "The theory of commuting Boolean sigma-algebras." Advances in Mathematics 144.1 (1999): 94-116. | |
| May 12, 2017 at 17:10 | comment | added | Henry.L | (3) I do not say a dual $\sigma$-algebra is necessary for Forman's computation, I just pointed out that it is unlike the case in Hilbert spaces where both Hilbert space and its dual are well-structured. We cannot treat homology as summary statistics in any sense, summary statistics is a random variable, which is already an element in the dual $\sigma$-algebra. | |
| May 12, 2017 at 17:08 | comment | added | Henry.L | It is aimed at addressing [Forman]. (1) As you cited, "any nontrivial homology class in $\tilde{H}_{\bullet}(\mathcal{J}_{\mathcal{X}})$ is an obstruction to evasiveness of statistical independence", would it be easier to calculate it directly or do it via cohomology? Also, it is not natural that we do not know anything about $\Lambda$. (2)Have you tried to calculate for one $\sigma$-algebra? For a bit more complicated, yet regular, $\sigma$-algebra like filtrations of brownian motion, such a computation becomes not tractable for me, @Chill2Macht | |
| May 12, 2017 at 17:04 | comment | added | Chill2Macht | Or is the answer not meant to address the entire question, but just the aspects which relate to Forman's paper? (That is sort of the impression I get from the original comment on the answer, but since the answer is very thorough and I don't really understand most of it, it is unclear to me how much/which aspects of the question it is intended to address, and which, if any, it is not.) | |
| May 12, 2017 at 17:02 | comment | added | Chill2Macht | Also, why do we need to consider multiple $\sigma$-(sub)algebras simultaneously? I was just thinking of calculating the homology of a given (single) $\sigma$-algebra, via the independence lattice, and trying to find a rule to calculate the same homology using arithmetic on marginal and joint distribution (functions), or arithmetic with entropy functionals, nothing too complicated. I.e. sort of using the homology as "summary statistics" for the independence structure of the RVs. I don't see yet how dual $\sigma$-algebras would be necessary to do this. | |
| May 12, 2017 at 16:57 | comment | added | Chill2Macht | What about the comment about needing a dual notion in order to use topology? Wouldn't we only need a dual notion to use cohomology, not homology? What would be wrong necessarily with only using homology? I want to accept this answer, but I'm really not sure at all if I understand the reasoning behind it. | |
| May 12, 2017 at 16:52 | comment | added | Henry.L | Here is a reference for axiomatic information theory in stats. Hope helps! stat.cmu.edu/~cshalizi/350/2008/lectures/06a/lecture-06a.pdf @Chill2Macht | |
| May 12, 2017 at 15:45 | comment | added | Chill2Macht | Thank you for the well-researched and knowledgeable answer, so much so that I am still working to understand it. Would you mind clarifying what is "axiomatic information theory" and the statement "If you do not even know the structure of this fundamental dual notion there is little hope that you can apply any topological trick", perhaps with references, if possible? I want to learn more information theory in the future, but right now my understanding of it is very basic (undegraduate level or below), so references for the background to understand some of the arguments used would be helpful. | |
| May 12, 2017 at 15:22 | history | edited | Henry.L | CC BY-SA 3.0 |
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| May 12, 2017 at 15:10 | comment | added | Henry.L | This aimed at explaining why there is not such a treatment existing according to [Forman]'s work as pointed out by OP. | |
| May 12, 2017 at 15:09 | history | answered | Henry.L | CC BY-SA 3.0 |