The probability of “undefined” (non-converging) output in generating the universal probability distribution

  • Authors:

  • Ray J. Solomonoff

  • Affiliations:

  • Computer Learning Research Centre, Royal Holloway, University of London, UK and IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland

  • Venue:
  • Information Processing Letters
  • Year:
  • 2008

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Abstract

In order to generate a universal probability distribution to extrapolate a binary string x of length i, we feed random bits into a universal device, M. When we find an input string that gives an output matching x, we continue the successful input with random bits until M produces a zero or one as output. The relative probabilities of these two continuations can give a normalized prediction for the probability of the symbol following x. There is, however, a probability, P"i"+"1(u) that the continued random input string will not generate any output for the (i+1)th symbol. We will show E"@m@?"i"="1^nP"i(u)=