information processing inequality

Also known as the data-processing inequality. Let $X\to Y\to Z$ form a Markov chain, i.e., $p(x,y,z)=p(x)p(y|x)p(z|x,y)$. Then the mutual information between $X$ and $Y$ cannot be increased by using $Z$ instead of $Y$:

$$I(X;Z)\le I(X;Y).$$

This is a fundamental inequality in information theory, telling us that any post-processing if $Y$ (resulting in $Z$), cannot increase the mutual information with the original source $X$. In other words, no transformation of the received code can give more information than the code itself.