# Data processing inequality for interaction information

The interaction information is defined as $$I(X;Y)-I(X;Y|Z)$$. Let $$Z-(X, Y) -(X', Y')$$ be a Markov chain. Is there an inequality similar to the data processing inequality, relating $$I(X';Y')-I(X';Y'|Z)$$ to $$I(X;Y)-I(X;Y|Z)$$? Thanks in advance.

Since the interaction information can be either negative or positive, and a Markov chain that erases everything can be used to take the interaction information to $$0$$, without further conditions the answer is "no".