Solomonoff Induction Violates Nicod's Criterion
This addresses a foundational issue in inductive reasoning and Bayesian inference for researchers in AI and philosophy, but it is incremental as it critiques and refines existing theoretical frameworks.
The paper tackles the problem of Solomonoff induction not satisfying Nicod's criterion, showing that observing black ravens can decrease belief in the hypothesis that all ravens are black, with differences between unnormalized and normalized priors leading to infinite or finite decreases in belief.
Nicod's criterion states that observing a black raven is evidence for the hypothesis H that all ravens are black. We show that Solomonoff induction does not satisfy Nicod's criterion: there are time steps in which observing black ravens decreases the belief in H. Moreover, while observing any computable infinite string compatible with H, the belief in H decreases infinitely often when using the unnormalized Solomonoff prior, but only finitely often when using the normalized Solomonoff prior. We argue that the fault is not with Solomonoff induction; instead we should reject Nicod's criterion.