The hypotheses in Solomonoff induction don’t binarily assert whether an output is allowed or not. What they do output might differ based on the details of your implementation, however.
The simplest approach would be to require the hypotheses to give a definite output. It’s true that then unpredictable or stochastic stuff cannot be fitted using a single hypothesis, but you can have an exponentially large family of hypotheses which are parameterized by a noise variable. There would then be some hypothesis within the family which matches the observations, and it would receive the probability.
Another option would be to allow the hypotheses to make probabilistic predictions, rather than deterministic predictions or logically binary nondeterministic predictions. In such a case, one would adjust the probability for each hypothesis continuously using Bayes’ theorem, rather than throwing out hypotheses.
Since any probabilistic hypothesis can be turned into a family of deterministic hypotheses augmented with a stream of noise, and since any deterministic hypothesis is also a probabilistic hypothesis of similar complexity, these two approaches will yield essentially the same results/predictions.
The hypotheses in Solomonoff induction don’t binarily assert whether an output is allowed or not. What they do output might differ based on the details of your implementation, however.
The simplest approach would be to require the hypotheses to give a definite output. It’s true that then unpredictable or stochastic stuff cannot be fitted using a single hypothesis, but you can have an exponentially large family of hypotheses which are parameterized by a noise variable. There would then be some hypothesis within the family which matches the observations, and it would receive the probability.
Another option would be to allow the hypotheses to make probabilistic predictions, rather than deterministic predictions or logically binary nondeterministic predictions. In such a case, one would adjust the probability for each hypothesis continuously using Bayes’ theorem, rather than throwing out hypotheses.
Since any probabilistic hypothesis can be turned into a family of deterministic hypotheses augmented with a stream of noise, and since any deterministic hypothesis is also a probabilistic hypothesis of similar complexity, these two approaches will yield essentially the same results/predictions.