The original Solomonoff induction is already equivalent to that.
Proof sketch: Any program that computes a probability distribution can (with constant overhead and a source of uniform random bits) be converted into a program that samples from that same distribution. Then convert that into a family of deterministic programs, each of which hardcodes one possible random sequence.
The total weight of such a family in the Solomonoff prior is within a constant factor of the weight that the single probabilistic hypothesis would have had, and ruling out members of the family that disagreed with observation is equivalent to Bayesian updating.
The original Solomonoff induction is already equivalent to that.
Proof sketch: Any program that computes a probability distribution can (with constant overhead and a source of uniform random bits) be converted into a program that samples from that same distribution. Then convert that into a family of deterministic programs, each of which hardcodes one possible random sequence.
The total weight of such a family in the Solomonoff prior is within a constant factor of the weight that the single probabilistic hypothesis would have had, and ruling out members of the family that disagreed with observation is equivalent to Bayesian updating.