Solomonoff Induction Violates Nicod's Criterion

TL;DR

This paper demonstrates that Solomonoff induction can violate Nicod's criterion by decreasing belief in a hypothesis upon observing confirming evidence, challenging traditional notions of evidence in inductive reasoning.

Contribution

It shows that Solomonoff induction does not satisfy Nicod's criterion and argues for rejecting Nicod's criterion in the context of Solomonoff induction.

Findings

Belief in H decreases infinitely often with unnormalized prior.

Belief in H decreases only finitely often with normalized prior.

Solomonoff induction can violate Nicod's criterion.

Abstract

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.