Thursday, February 3, 2011

Pigeonholes and entropy, a work in progress

The pigeonhole principle is probably the simplest lemma in combinatorics:
• Say you want to sort k objects into n bins.  If n < k, then at least one bin must contain at least two objects.
For example, a person can have only one of 366 possible birthdays (counting Feb 29).  By the pigeonhole principle, any gathering of 367 or more people must have people who share a birthday.

The lemma is obvious almost to the point of triviality, so it comes as a surprise that you can use it to prove powerful results; there are some examples in the book.

Lately I've been wondering whether it can be used to say anything interesting about entropy.  Here's the first thing I thought of: let X and Y be random variables on a discrete alphabet, and let S(X) and S(Y) represent the support of the probabilities of X and Y, respectively (i.e., x is in S(X) if and only if p(x) > 0). Then
• Theorem. If |S(X)| > |S(Y)|, then H(X|Y) > 0.
• Proof. By definition of entropy, H(X|Y) >= 0. If H(X|Y) = 0, then there must exist an injective map from S(X) --> S(Y).  However, since |S(X)| > |S(Y)|, no such map exists (by the pigeonhole principle).  Thus, H(X|Y) != 0, and the theorem follows.
Not very exciting, I will admit.  I'm still thinking about it, any other ideas?