Raymond Yeung

Information Inequalities, Conditional Independence, and Groups

In this talk, we will first describe a geometrical framework for information inequalities, which is now known to be related not only to information theory, but also to probability theory, group theory, Kolmogorov complexity, and possibly quantum information theory and thermodynamics. From this framework, we can see that the studies of conditional independence in probability theory is a special problem in the studies of information inequalities. We will then discuss a one-to-one correspondence between information inequalities and a type of inequalities in group theory. This one-to-one correspondence establishes a bridge between information theory and group theory, which are two seemingly unrelated subjects. The major implication of this result is that we can prove information inequalities by proving the corresponding group inequalities, and vice versa. By giving a group-theoretic proof for all so-called Shannon-type information inequalities, we suggest that new inequalities can be discovered by making use of the rich set of tools in group theory. On the other hand, via a non-Shannon-type information inequality recently discovered by Zhang and Yeung, we obtain a new inequality in group theory whose meaning is yet to be understood.