Additive number theory contains a number of so-called "sumset inequalities" that relate the cardinalities of various finite subsets of an abelian group G, for instance, the sumset A+A and the difference set A-A of a finite subset A of G. It also contains "inverse" results such as Freiman's theorem, which asserts that sets A such that A+A is relatively small must have some "additive structure". Motivated by considerations coming from multiple directions including probability theory, combinatorics, information theory, and convex geometry, we explore probabilistic analogues of such results in the general setting of locally compact abelian groups. For instance, we show that for independent, identically distributed random variables X and X' whose distribution has a density with respect to Haar measure on a locally compact abelian group G, the entropies of X+X' and X-X' strongly constrain each other. We will also discuss stronger statements that can be made for specific groups of interest, such as R^n, the integers, and finite cyclic groups. Based on (multiple) joint works with Ioannis Kontoyiannis (Athens), Jiange Li (Delaware), Liyao Wang (Yale), and Jaeoh Woo (Yale).