conditional mutual information and how to deal with zero probabilities

Question

the conditional mutual between three sets of mutually exclusive variables, X, Y, and Z, is defined as follows.

$I(X,Y|Z) = \sum_{xyz} P(x,y,z) \log \frac{P(z)P(x,y,z)}{P(x,z)P(y,z)}$

my questions concern the $\log$ of the ratio of the probability products.

1. if $P(z)$ or $P(x,y,z)$ is 0, then $\log(0)$ is undefined.
2. if $P(x,z)$ or $P(y,z)$ is 0, then $\log(\infty)$ is undefined.

how do i deal with these 2 situations? the approach can be very flexible. for example, i thought about ignoring the inner sums where such conditions occur, but is this correct or reasonable?

any help is appreciated.

Answer 1

Ignoring the terms where this happens is the correct thing to do. You can justify this by noting that in each case you've outlined, no matter what happens inside the $\log$ you will have $P(x,y,z) = 0$. You can see this by applying the Frechet inequalities, namely that $P(A,B) \le \min\{P(A), P(B)\}$.