  We recall some of the history of the information-theoretic approach to
deriving core results in probability theory and indicate parts of the recent
resurgence of interest in this area with current progress along several
interesting directions. Then we give a new information-theoretic proof of a
finite version of de Finetti's classical representation theorem for
finite-valued random variables. We derive an upper bound on the relative
entropy between the distribution of the first $k$ in a sequence of $n$
exchangeable random variables, and an appropriate mixture over product
distributions. The mixing measure is characterised as the law of the empirical
measure of the original sequence, and de Finetti's result is recovered as a
corollary. The proof is nicely motivated by the Gibbs conditioning principle in
connection with statistical mechanics, and it follows along an appealing
sequence of steps. The technical estimates required for these steps are
obtained via the use of a collection of combinatorial tools known within
information theory as `the method of types.'
