Leike & Hutter (2015) proved Solomonoff induction violates Nicod's confirmation criterion

confirmation-theoryalgorithmic-information-theorysolomonoff-inductionAIXIjean-nicodcross-domain-bridgecross-time-bridge

Jan Leike and Marcus Hutter (Australian National University), working within the theory behind Hutter's AIXI model of universal artificial intelligence, proved in 2015 that Solomonoff induction — the incomputable ideal Bayesian predictor underlying AIXI — fails Jean Nicod's 1930 confirmation criterion: "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." Under the unnormalized Solomonoff prior this failure recurs infinitely often for some computable sequences; switching to the normalized prior bounds the violations to finitely many, but does not eliminate them.

The authors' own diagnosis runs against intuition: rather than treat the result as a defect in Solomonoff induction, they conclude that Nicod's criterion itself should be rejected as a general principle of confirmation. Solomonoff induction is otherwise established, elsewhere in the algorithmic-information-theory literature, as asymptotically optimal at sequence prediction — so a formal predictor that is provably near-optimal overall can still, at specific finite time steps, have a confirming observation lower rather than raise its belief in a hypothesis. That gap is the paper's case that the 85-year-old criterion, not the induction method, is what fails to hold up.