PFR

5 The Fibring lemma

Proposition 5.1 General fibring identity

Let π:HH be a homomorphism additive groups, and let Z1,Z2 be H-valued random variables. Then we have

d[Z1;Z2]d[π(Z1);π(Z2)]+d[Z1|π(Z1);Z2|π(Z2)].

Moreover, if Z1,Z2 are taken to be independent, then the difference between the two sides is

I(Z1Z2:(π(Z1),π(Z2))|π(Z1Z2)).
Proof
Corollary 5.2
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If π:GH is a homomorphism of additive groups and X,Y are G-valued random variables then

d[X;Y]d[π(X);π(Y)].
Proof
Corollary 5.3 Specific fibring identity
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Let Y1,Y2,Y3 and Y4 be independent G-valued random variables. Then

d[Y1+Y3;Y2+Y4]+d[Y1|Y1+Y3;Y2|Y2+Y4]+I[Y1+Y2:Y2+Y4|Y1+Y2+Y3+Y4]=d[Y1;Y2]+d[Y3;Y4].
Proof