The Existence of Predictive Complexity and the Legendre Transformation

Predictive complexity is a generalisation of Kolmogorov complexity. In this paper we point out some properties of predictive complexity connected with the Legendre ({Young{Fenchel) transformation. Our main result is that mixability is necessary for the existence of conditional predictive complexity (it is known to be su cient under very mild assumptions). We formulate a di erential criterion of mixability and show that it reduces to a very simple form if we employ the Legendre transformation. The Legendre transformation also turns out to have a probabilistic meaning which allows us to prove that a variant of predictive complexity speci es a unique (up to a parametrisation) mixable game.