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let $Y_1 = X_1 + X_2 + X_3 + X_4 $ and $Y_2 = X_1 + X_2 + X_5 + X_6 $

where $X ~ iid $ Binary valued RVs

I want to estimate $X_1$ and $X_2$. How can calculating $I(Y_1;Y_2)=H(Y_1)-H(Y_1/Y_2)$ help me in this problem. The conditional entropy $H(Y_1/Y_2)$ reduces uncertainty about $Y_1$ but we know that $Y_1,Y_2$ are dependent variables through $X_1,X_2$ so using $H(Y_1/Y_2)$ should be helpful in better estimating $X_1,X_2$

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What do you mean by "estimate $X_1$ and $X_1$"? Do you only know $Y_1$ and $Y_2$ (in which case you can only estimate up to $X_1 + X_2$)? – Memming Jan 31 '13 at 0:45

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