7 - Biased algorithmic randomness

Summary

In this survey, we lay out the central results in the study of algorithmic randomness with respect to biased probability measures. The first part of the survey covers biased randomness with respect to computable measures. The central technique in this area is the transformation of random sequences via certain randomness-preserving Turing functionals, which can be used to induce non-uniform probability measures. The second part of the survey covers biased randomness with respect to non-computable measures, with an emphasis on the work of Reimann and Slaman on the topic, as well as the contributions of Miller and Day in developing Levin's notion of a neutral measure. We also discuss blind randomness as well as van Lambalgen's theorem for both computable and non-computable measures. As there is no currently-available source covering all of these topics, this survey fills a notable gap in the algorithmic randomness literature.