Cameron Gordon

# Telephone and the Data Processing Inequality

A basic rule of mathematics is that it should seek to illuminate, not obscure. A complicated concept should be understandable by a grandmother or a preschooler. In this spirit here is a primer on one of the most powerful concepts in information theory - the data processing inequality - using the __game of Telephone__.

In Telephone a message is transmitted (e.g. "This is a farcical example") through a line of people, each relying on their best interpretation of the whisper of the person before them. After a few people, the message begins to get garbled "This is a fart icicle sample", and before long the message turns to complete nonsense.*

Why does this happen? It turns out that we can explain this through the data processing inequality. In short: information retained from the original message can only decrease as the number of people increases.

A line of people is a * Markov chain* - a series of nodes in which each node's state depends only on the node directly before it. The message can be described by the

*information*it contains. The information that any two nodes contain that is the same is known as their

*mutual information*.

As each node depends only on the previous for the message (which is transmits as faithfully as it can) information can be lost. The mutual information between person 1 and person N is *less than or equal* to that between person 1 and 2.

This is the data processing inequality. More formally, for a Markov Chain X -> Y -> Z and denoting the mutual information be two nodes A, B as I(A; B), we have I(X;Y) ≥ I(X;Z).

The data processing inequality is extremely useful. In telecommunications it describes how signal degrades over a transmission line and requires boosting. In business it show why going direct to the source is often more useful than dealing with a line of intermediaries. In neural networks, it helps explain the __vanishing gradient problem__ in very deep networks. And in organisational structures - a topic I'll return to in a later post - it explains why important information can be lost between ground level staff and board-level management.

* For reasons unknown to humanity with a long enough line the final message will invariably end with the phrase "Purple monkey dishwasher".