Solomonoff's theory of inductive inference
Ray Solomonoff's theory of universal inductive inference is a theory of prediction based on logical observations, such as predicting the next symbol based upon a given series of symbols. The only assumption that the theory makes is that the environment follows some unknown but computable probability distribution. It is a mathematical formalization of Occam's razor and the Principle of Multiple Explanations.
Prediction is done using a completely Bayesian framework. The universal prior is taken over the class of all computable sequences—this is the universal a priori probability distribution; no computable hypothesis will have a zero probability. This means that Bayes rule of causation can be used in predicting the continuation of any particular computable sequence.
The theory is based in philosophical foundations, and was founded by Ray Solomonoff around 1960. It is a mathematically formalized combination of Occam's razor and the Principle of Multiple Explanations. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.
The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to one (as per the definition of probability) thus the probabilities must roughly decrease as we enumerate the infinite set of all programs, otherwise S will be strictly greater than one. To be more precise, for every > 0, there is some length l such that the probability of all programs longer than l is at most . This does not, however, preclude very long programs from having very high probability.
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