-.- Two expected causal information measures CopyRight (C) 2007, Jan Hajek, NL, version 1.0 of 2007-10-22 NO part of this document may be published, implemented, programmed, copied or communicated by any means without an explicit & full reference to this author + the full title + the website www.humintel.com/hajek for the freshest version + the CopyRight note in texts and in ALL references to this. An implicit, incomplete, indirect, disconnected or unlinked reference (in your text and/or on www) does NOT suffice. All based on 1st-hand experience. ALL rights reserved. In my other epapers on causation at www.humintel.com/hajek I explain and analyze in great detail the Relative difference RD or RDS by Mindel Sheps (Harvard 1959), reinvented by Frank Restle (1961), by John Swets (1964), and by Patricia Cheng (1997). Let x be an event assumed to cause the occurence of an effect-event y ; ie x, y are (quasi)random events. Let X be a set of various x's , and Y be a set of various y's ; ie X, Y are compound (quasi)random variables aka r.v.'s. ARR(y:x) = P(y|x) - P(y|~x) = the well known Absolute error reduction . IF ARR(y:x) >= 0 THEN RD(y:x) = ARR(y:x)/(1 - P(y|~x)) >=0 ie y is said to be generated by x ELSE RD(y:x) = ARR(y:x)/(1 - P(y|x) ) < 0 ie y is said to be prevented by x, please find HF in my other epapers on causation at www.humintel.com/hajek . Based on this RD(y:x) understood as a measure of how much an event x is (anti)causing an event y , I propose here 2 Expected causal information measures EciS and EciA defined thus : EciS(Y:X) = Expected Signed causal information on Y caused by X = E[ RD(y:x) ] = Sum_y[ Sum_x[ RD(y:x).P(x,y) ] ] where the . is a multiplication. EciA(Y:X) = Expected Absolute causal information on Y caused by X = E[ Abs(RD(y:x)) ] = Sum_y[ Sum_x[ Abs(RD(y:x)).P(x,y) ] ] EciS's properties : Eci stands for both EciA as well as for EciS . Eci(Y:X) <> Eci(X:Y) in general, which is a must for a measure of causation. 0 <= I(Y:X) = I(X:Y) holds for Shannon's mutual information I(:) which has a coding interpretation, which is irrelevant to causation. The symmetry of I(:) wrt X, Y makes it UNsuitable as a measure of causation, although I(:) may be used as a measure of mutual association which, due to the adjective "mutual", is symmetrical wrt X, Y. 0 <= Abs(Eci(Y:X)) <= 1 follows from the fact that 0 <= abs(RD(y:x)) <= 1, and from Sum_y[ Sum_x[ P(x,y) ] ] = 1. 0 <= EciA <= 1 ; -1 <= EciS <= 1 , the sign of EciS carries information not available from I(Y:X) or EciA EciS(Y:X) > 0 indicates Expected generativity of the effect variable Y due to the causation of Y by X ; EciS(Y:X) < 0 indicates Expected preventativity of the effect variable Y due to the hindrance, blocking or prevention of Y by X . Appendix (taken from my epapers at www.humintel.com/hajek ) : Absolute risk reduction ARR = slope of a probabilistic regression line for the occurence of an event y assumed to be dependent on the occurrence of an event x : ARR(~y:~x) = P(~y|~x) - P(~y|x) = [ 1 - P(y|~x) ] - [ 1 - P(y|x) ] = ARR( y: x) = P(y|x) - P(y|~x) = the standard absolute risk reduction = [ Pxy - Px.Py ]/[ Px.(1-Px) ] = cov(x,y)/var(x) = beta(y:x) = the slope of a probabilistic regression line Py = beta(y:x).Px + alpha(y:x) from which follows my fresh interpretation of GF and my HF . References : Abundant references are in my other epapers on causation at www.humintel.com/hajek ; search my epapers for GF , HF , RDS( , RD( -.-