KLD is a measure  of the non-symmetric difference between two probability distributions P and Q. While MI is defined as Kullback-Leiber distance between joint probability distribution of two random variables and product of their marginal distribution.

Mutual information between X and Y can be thought of as a reduction in the uncertainty of X given Y,i.e. I(X,Y)= H(X)-H(X/Y)

it can be related to KL divergence (note that KL is not a norm as it is non symmetric) as follows (and as mentioned above).

KL between the joint distribution and the product of the individual distributions.

Intuitively, if X, Y are independent, p(x, y) = p(x)p(y), and the KL-divergence is zero, and knowing X gives zero information gain about Y

see this:

http://pages.cs.wisc.edu/~jerryzhu/cs769/info.pdf

Please how we can categorize KL and MI ?

Can we say/;

MI related to Information-based measures

KL related to  Probability-based measures

??

Mutual information is a symmetric measure, i.e. I(X;Y)=I(Y;X).

However KL-Distance is an asymmetric measure: D(P||Q) not equal D(Q||P). There is a few symmetric form of KL-Distance such as KL-Symmetric, D(P||Q)+D(Q||P), and Jensen-Shannon measure.

I(x;y)=KL(p(x,y)||p(x)p(y)) = H(x)-H(x|y)=H(y)-H(y|x)

Also:

If x & y are independent, then mutual information between x & y = I(x;y) = 0 and p(x,y) = p(x)p(y) .

The more independent x & y are, the closer p(x,y) and p(x)p(y) are.

If x & y are independent and the KL-divergence is zero, then knowing x gives zero information gain about y and knowing y gives zero information gain about x.

I(x;y) measures the reduction of uncertainty of x (or y) due to knowing y (or x).

H(y|x), the conditional entropy, measures the uncertainty of y given that we know x.

H(x|y) measure the uncertainty of x given that we know y.