A powerful technique to quantify the trend to equilibrium and the best constants in the associated functional inequalities for diffusions on Riemannian manifolds consists in establishing convexity estimates for the relative entropy via the so called Gamma calculus. In order to adapt these ideas to the context of discrete Markov chains several notions of discrete curvature have been recently introduced and used to obtain concrete lower bounds for the logarithmic Sobolev constant in a number of situations. However, the picture is not fully clear yet and several natural questions remain unanswered. In this talk, I will present a more probabilistic approach to convex entropy decay which relies on the notion of coupling rates to bypass or replace discrete Böchner identities. If time allows, I will show how this approach produces explicit lower bounds in non perturbative setting, thus going beyond the weak-interaction/low high temperature regime.