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What can we learn from information?

In this talk I will discuss some of the basics of information in probability and statistics, and how, both intuitively and theoretically, information and information differences can be used to represent statistical dependence. I will cover both the "information theory" perspective, and also an underappreciated quantity called the Density Information Matrix (DIM). Despite being more than a decade old, and having some remarkable theoretical properties, the Density Information Matrix is almost completely unknown in the statistics community. I introduce some of these remarkable properties, and also discuss some of the practical difficulties in estimating the DIM, before covering some work in progress for improving its estimation.