Data processing inequality

The Data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical operation. This can be expressed concisely as 'post-processing cannot increase information'. As explained by Kinney and Atwal, the DPI means that information is generally lost (never gained) when transmitted through a noisy channel.

Definition

Let three random variables form the Markov chain $X\rightarrow Y\rightarrow Z$, implying that the conditional distribution of $Z$ depends only on $Y$ and is conditionally independent of $X$. Specifically, we have such a Markov chain if the joint probability mass function can be written as

$p(x,y,z)=p(x)p(y|x)p(z|y)$

In this setting, no processing of Y , deterministic or random, can increase the information that Y contains about X. Using the mutual information, this can be written as :

$I(x;y)\geqslant I(x;z)$

With the equality $I(x;y)=I(x;z)$ if and only if $X\rightarrow Z\rightarrow Y$.

See also

• Garbage in, garbage out

External links

• http://www.scholarpedia.org/article/Mutual_information