Data processing inequality

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The data processing inequality is an information theoretic concept that 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'.[1]


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

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

with the equality if and only if . That is, and contain the same information about , and also forms a Markov chain.[2]


One can apply the chain rule for mutual information to obtain two different decompositions of :

By the relationship , we know that and are conditionally independent, given , which means the conditional mutual information, . The data processing inequality then follows from the non-negativity of .

See also[edit]


  1. ^ Beaudry, Normand (2012), "An intuitive proof of the data processing inequality", Quantum Information & Computation, 12 (5–6): 432–441, arXiv:1107.0740, Bibcode:2011arXiv1107.0740B, doi:10.26421/QIC12.5-6-4, S2CID 9531510
  2. ^ Cover; Thomas (2012). Elements of information theory. John Wiley & Sons.

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