In this talk, I explore a few instances of the application of strong versions of data processing inequalities to different estimation problems where the estimation procedures are restricted in some way. In particular, I will survey several recent results, with applications in (1) private statistical inference, (2) large-scale distributed estimation, and (3) stochastic optimization, all of which rely on a few new—but related—information theoretic tools that help us to understand the fundamental limits of, and optimal algorithms for, a variety of inference problems.