Three Kinds of Probabilistic Induction: Universal Distributions and Convergence Theorems

We will describe three kinds of probabilistic induction problems, and give general solutions for each , with associated convergence theorems that show they tend to give good probability estimates. The first kind extrapolates a sequence of strings and/or numbers. The second extrapolates an unordered set of strings and/or numbers. The third extrapolates an unordered set of ordered pairs of elements that may be strings and/or numbers. Given the first element of a new pair, to get a probability distribution on possible second elements of the pair. Each of the three kinds of problems is solved using an associated universal distribution. In each case a corresponding convergence theorem is given, showing that as sample size grows, the expected error in probability estimate decreases rapidly. The solutions given are very general and cover a great variety of induction problems. Time series prediction, grammar discovery (for both formal and natural languages), curve fitting, the identification ...

• Type: Journal
• Year: 2008
• Where: CJ
• Authors: Ray J. Solomonoff
• Keywords: CJ 2008, Induction Problems, Probabilistic Induction Problems, Unordered Set