# Monthly Archives: October 2013

## An optimal superfarthingale and its convergence over a computable topological space

News
Oct, 2013. Online
Sep 3, 2011. Accepted
June 7, 2011. Submitted

Title
An optimal superfarthingale and its convergence over a computable topological space

Type
Conference paper

Conference
Solomonoff 85th Memorial Conference at Melbourne, Australia.

Lecture Notes in Computer Science
Volume 7070 2013
Algorithmic Probability and Friends. Bayesian Prediction and Artificial Intelligence
Papers from the Ray Solomonoff 85th Memorial Conference, Melbourne, VIC, Australia, November 30 – December 2, 2011

preprint

Abstract
We generalize the convergenece of an optimal semimeasure
to a real probability in algorithmic probability by using game-theoretic
probability theory and the theory of computable topology. Two lemmas
in the proof give as corollary the existence of an optimal test and an
optimal integral test, which are important from the point of view of
algorithmic randomness. We only consider an SCT3 space, where we
can approximate the measure of an open set. Our proof of almost-sure
convergence to the real probability by a superfarthingale indicates why
the convergence in Martin-L¨of sense does not hold.