  In this paper, we propose a generalization of Bakry-\'Emery's calculus which
allows us to formulate both Bakry-\'Emery and entropic curvature
simultaneously. This formulation represents both curvatures as an integral of
the Bochner formula against some measure. This leads to a natural optimality
criterion of measures, which we investigate in the Bakry-\'Emery setting.
Moreover, our approach leads also to a dimension parameter in the framework of
entropic curvature.
  We also present gradient estimate applications, that is, diameter estimates
for Markov chains with strictly positive entropic curvature and a spectral gap
estimate. The latter implies non-existence of non-negatively curved expanders.
