Is this conditional mutual information relationship valid

Question

We have five random variables $N_1,N_2$ and $Z_1, Z_2,Z_3$, such that

1. $N_1$ and $N_2$ are mutually independent,
2. $Z_1$ and $Z_2$ are mutually independent,
3. $N_1$ and $Z_2$ are mutually independent,
4. $N_2$ and $Z_1$ are mutually independent,
5. $Z_1$ and $Z_2$ are conditionally independent given $Z_3$.

Given above conditions, is the following mutual information relationship valid:

$I(N_1\: \: N_2 \:;\: Z_3 \: \Big| \: Z_1 \: \: Z_2)=I(N_1 \:;\: Z_3 \: \Big| \: Z_1 )+I(N_2 \:;\: Z_3 \: \Big| \: Z_2).$

Answer 1

Chain rule gives you (there is another way of applying chain rule here but same steps can be applied to it.): $$ I(N_1\: \: N_2 \:;\: Z_3 \: \Big| \: Z_1 \: \: Z_2)=I(N_1 \:;\: Z_3 \: \Big| \: Z_1,Z_2 )+I(N_2 \:;\: Z_3 \: \Big| \: Z_2,Z_1,N_1). $$ To have the proposed equality, you need: $$ I(N_1 \:;\: Z_3 \: \Big| \: Z_1,Z_2 )= I(N_1 \:;\: Z_3 \: \Big| \: Z_1) $$ and $$ I(N_2 \:;\: Z_3 \: \Big| \: Z_2,Z_1,N_1)=I(N_2 \:;\: Z_3 \: \Big| \: Z_2). $$ In general, this is not valid. Take $Z_3=(W_1,W_2)$ and $Z_1=N_1+W_1$ and $Z_2=N_2+W_2$ and $W_1,W_2,N_1,N_2$ are independent. You can check that the above equalities will not hold. For instance: $$ I(N_1 \:;\: Z_3 \: \Big| \: Z_1,Z_2 )=I(N_1 \:;\: W_1,W_2 \: \Big| \: W_1+N_1,W_2+N_2 )\\ \neq\\ I(N_1 \:;\: Z_3 \: \Big| \: Z_1)=I(N_1 \:;\: W_1,W_2 \: \Big| \: W_1+N_1). $$ Intuitively $Z_2$ and $Z_1$ are not independent of $Z_3$ and therefore they can contain information about it, so removing them from mutual information can change its value.