In two influential papers written in the 1950's, David Blackwell asked the following question: if an agent wants to make an optimal decision about some "state of Nature" and can choose between two observation channels (or statistical experiments), which one should he prefer? Blackwell has shown that the decision-theoretic criterion is equivalent to a less informative experiment being stochastically degraded with respect to a more informative one. This induces a partial ordering on the set of all observation channels with the same input alphabet; alas, most channels are incomparable.

The scope of Blackwell's theory was expanded considerably by Lucien Le Cam, who showed that to any two observation channels with the same input alphabet one can associate a number between 0 and 1, called the "deficiency," which quantifies the extent to which one channel can be stochastically degraded to resemble the other in the total variation distance. Le Cam's theory, while possessing a great deal of abstraction, has since then been embraced by the mathematical statistics community because it offered a principled way of approximating "complicated" statistical experiments by "simpler" ones.

In this talk, I will present an application of the framework of Blackwell and Le Cam to the problem of channel coding in information theory. Since the cascade of a code and a channel is a statistical experiment, we can endow Blackwell's ordering, as well as the Le Cam's deficiency measure, with an operational significance, pertaining to how much we expect to lose if we use a code designed for one channel on another channel. This leads to a number of useful upper and lower bounds on the maximal probabilities of error for channel codes. I will compare the Blackwell ordering with another ordering of channels, due to Shannon and show that the latter is weaker than the former. Finally, I will present a generalization of Le Cam deficiency distance using more general divergence measures between probability distributions that have a monotonicity property with respect to data processing (the so-called g-divergences introduced recently by Polyanskiy and Verdu).