On-Line Probability, Complexity and Randomness

Abstract. Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned. We adapt the notions of algorithmic information theory (complexity, algorithmic randomness, martingales, a priori probability) to this framework and show that many classical results are still valid. 1 On-line probability distributions Consider the following “real-life” situation. There is a tournament (say, chess or football); before each game the referee tosses a coin to decide which player will start the next game. Assuming the referee is honest, we would be surprised to learn that, say, all 100 coin tosses have produced a tail. We would be surprised also if the result of the coin tossing always turned out to be equal to som...

• Type: Conference
• Year: 2008
• Where: ALT
• Authors: Alexey V. Chernov, Alexander Shen, Nikolai K. Vereshchagin, Vladimir Vovk
• Keywords: ALT 2008, Coin Tossing, Machine Learning, On-line Probability Distributions, Theory Considers Probability