# How can we determine Conditional Mutual Information based on multiple conditions

Let X,Y,Z be jointly distributed, the conditional mutual information is defined as :-

Similarly if we have 4 (or more variables) say X,Y,Z,W then how can we determine the conditional mutual information between X and Y given Z and W. So basically, how to determine conditional mutual information between 2 variables based on multiple conditions.

• Where is this screenshot from? Commented Jan 29, 2023 at 23:07

Define the random variable $$V \triangleq (W, Z)$$ and this becomes a case of the equation in your screenshot. You can then compute $$\operatorname{I}(X; Y \mid V)$$.

• So you mean V is the joint probablity of the multiple conditions. In this case it will be joint probablity between W and Z. Could you please elaborate if I am getting it right.
– Abhi
Commented May 17, 2021 at 20:27
• Yeah, that's right. Commented May 17, 2021 at 20:29
• Thanks. Just one clarification, since I m not that familiar, if I want to calculate the joint probablity between 2 discreate random variables, X and Y. And X has 4 events(x1,x2,x3,x4) and Y has 2 (y1,y2). So the combined joint probablity will be the summation of all the events taken in different combinations, right. Like sum of joint prob of (x1,y1); (x1,y2); (x2,y1); (x2,y2)...
– Abhi
Commented May 18, 2021 at 10:05
• Also, can we extend the formula of conditional mutual information shown above to include more conditions using Entropy. Like instead of Z can we write H( XWZ) + H(YWZ) + H(XYWZ) -H(WZ). Will such formula be applied ?
– Abhi
Commented May 18, 2021 at 12:06
• Yes, you’re right about the joint distribution. Yes, you can modify the formula like that. Commented May 18, 2021 at 14:12