Universal Artificial Intelligence by Margus Hutter provides a full formal approximation of Solomonoff Induction
Use randomness and select a subset of all hypotheses. i.e. Monte Carlo method. Perhaps use an evolutionary algorithm that that starts with seed hypotheses and vary the ones that get closest to the observation
Jurgen Schmidhuber equates faster algorithms with high probability, and uses that to create a very fast approximation. But the assumption that faster algorithms are more likely isn’t a given.