C.H.~Bennett, P.~G\'acs, M.~Li, P.M.B.~Vit\'anyi, and W.H.~Zurek have defined information distance between two strings $x$, $y$ as $$ d(x,y)=\max\{ K(x|y), K(y|x) \} $$ where $K(x|y)$ is the conditional Kolmogorov complexity. It is easy to see that for any string $x$ and any integer $n$ there is a string $y$ such that ... more >>>

A new notion of predictive complexity and corresponding amount of information are considered. Predictive complexity is a generalization of Kolmogorov complexity which bounds the ability of any algorithm to predict elements of a sequence of outcomes. We consider predictive complexity for a wide class of bounded loss functions which are ... more >>>