The Existence of Predictive Complexity and the Legendre Transformation
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by Yuri Kalnishkan, Volodya Vovk
http://www.clrc.rhbnc.ac.uk/tech-report/files/tr0004.ps
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Abstract:
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 sufficient under very mild assumptions). We formulate a differential 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 specifies a unique (up to a
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