Error in Enumerable Sequence Prediction

We outline a method for quantifying the error of a sequence prediction. With sequence predictions represented by semimeasures ν(x) we define their error to be − log2 ν(x). We note that enumerable semimeasures are those which model the sequence as the output of a computable system given unknown input. Using this we define the simulation complexity of a computable system C relative to another U giving an exact bound on their difference in error. This error in turn gives an exact upper bound on the number of predictions ν gets incorrect. 1 A prediction’s error Suppose we wish to predict a sequence over a finite alphabet X. Definition 1. [3] A semimeasure ν is a function ν:X∗ → [0, 1] satisfying: 1. Normalisation: ν( ) = 1 2. Coherence: ∑