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The common information of two dependent random variables

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1 Author(s)

The problem of finding a meaningful measure of the "common information" or "common randomness' of two discrete dependent random variables X,Y is studied. The quantity C(X; Y) is defined as the minimum possible value of I(X, Y; W) where the minimum is taken over all distributions defining an auxiliary random variable W \in mathcal{W} , a finite set, such that X, Y are conditionally independent given W . The main result of the paper is contained in two theorems which show that C(X; Y) is i) the minimum R_0 such that a sequence of independent copies of (X,Y) can be efficiently encoded into three binary streams W_0, W_1,W_2 with rates R_0,R_1,R_2 , respectively, [\sum R_i = H(X, Y)] and X recovered from (W_0, W_1) , and Y recovered from (W_0, W_2) , i.e., W_0 is the common stream; ii) the minimum binary rate R of the common input to independent processors that generate an approximation to X,Y .

Published in:

IEEE Transactions on Information Theory  (Volume:21 ,  Issue: 2 )