Monge-Ampère operators play an important role in the construction of invariant valuations on convex functions. Conversely, translation equivariant Monge-Ampère operators can be identified with certain measure-valued valuations on the space of finite convex functions. In this talk we will discuss how this description gives rise to additive kinematic formulas encoded in a convolution product. Moreover, we will see that this product structure satisfies the mixed Hodge-Riemann bilinear relations, which provides a natural way to interpret the higher order versions of Alexandrov’s mixed discriminant inequality.