I am trying to find some data structures/mathemetical theories to represent causal relationships which differ from graphical models or Bayesian Networks. Any ideas?
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$\begingroup$ There may be some usefule information found at an earlier question (which generated quite a bit of debate): mathoverflow.net/questions/22490/… $\endgroup$– Harald Hanche-OlsenCommented Jun 15, 2010 at 11:52
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$\begingroup$ I would say that most of it can be cast in terms of adapted filtrations. $\endgroup$– Grant IzmirlianCommented Jan 7, 2014 at 21:20
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3 Answers
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The standard account of causality is Lewis's theory of counterfactuals. He wrote a small, very readable book called Counterfactuals, which the SEP summarizes here. The idea is to take the viewpoint of modal logic, and interpret the counterfactual conditional $A \;\boxdot\!\!\!\to B$ (read "if A were the case, then B would be the case") as holding if $B$ holds in the nearest world in which $A$ holds. Obviously this requires enriching modal logic with a notion of similarity of worlds, as well.
A lot of other people have worked on this subject; personally I am most fond of McCain-Turner causality, since Graham White has given it a nice proof-theoretic formulation.
answered Jun 15, 2010 at 8:11
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2$\begingroup$ I wouldn't say there's a single standard account of causality. For a discussion of further accounts - mechanistic, probabilistic, agency - try kent.ac.uk/secl/philosophy/jw/2004/causality.pdf. $\endgroup$ Commented Jun 15, 2010 at 9:00
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$\begingroup$ That's a fair point -- I meant "standard" in the sense of "the one everyone feels obliged to argue with", rather than "the one everyone accepts". $\endgroup$ Commented Jun 15, 2010 at 9:41
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With several variables connected by asymmetric causal relations, it's not so likely that a mathematical theory of causality will escape graphical representation. Neel mentioned above Lewis's counterfactual analysis, and this has a close affinity with aspects of Judea Pearl's work on casual Bayesian networks, see p. 239 of http://books.google.co.uk/books?id=wnGU_TsW3BQC.
Among quantitative approaches, a variant on the usual statistical approach is Janzing and Schoelkopf's use of algorithmic dependence to determine causal relations: http://www.kyb.mpg.de/publications/attachments/paper_IEEE_version3_webseite_6526%5B1%5D.pdf.
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Burks, A.W. (1977), Chance, Cause, Reason : An Inquiry into the Nature of Scientific Evidence, University of Chicago Press, Chicago, IL.
