Solomonoff Induction Violates Nicods Criterion

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.

💡 Research Summary

The paper investigates whether Solomonoff induction, the formal Bayesian solution to the problem of induction, satisfies Nicod’s criterion—a classic principle stating that observing an instance of a predicate F that is also an instance of a predicate G (e.g., a black raven) should increase belief in the universal hypothesis “All Fs are Gs.” The authors formalize the hypothesis H (“all ravens are black”) using an alphabet of four symbols representing the four possible observations (black raven, non‑black raven, black non‑raven, non‑black non‑raven). They define the Solomonoff prior M as the probability that a universal monotone Turing machine outputs a given string when fed with random bits, and also consider its normalized version M_norm, which turns the semimeasure M into a proper probability measure.

The paper’s core technical contribution is a decomposition of all programs contributing to the prior into five categories (A–E): programs that are compatible with H but are falsified at the current step, programs incompatible with H that are falsified, programs compatible with H that correctly predict the current observation, programs incompatible with H that correctly predict, and programs that halt (enter an infinite loop). The contributions of these categories to the prior are denoted A, B, C, D, and E respectively. Lemma 4 shows that the observation at time t confirms H if and only if B·C > A·D + D·E; it disconfirms H when the opposite inequality holds. This simple algebraic condition makes it possible to analyze the dynamics of the posterior probability M(H | x₁:t).

Using bounds from algorithmic information theory (Kolmogorov complexity K(t) and its monotone lower bound m(t)), Lemma 6 establishes that A and B are on the order of 2^{−K(t)}, C is at least a constant, D is on the order of 2^{−m(t)} and tends to zero, and E also tends to zero as t grows. Substituting these bounds into Lemma 4 yields that for sufficiently large t the inequality B·C < A·D + D·E holds, meaning the posterior probability of H actually decreases after observing a black raven. This demonstrates a direct violation of Nicod’s criterion.

The authors construct an explicit example (Example 3) by mixing a specially designed semimeasure ρ with the universal prior M to obtain a new prior ξ = ρ + εM. With a suitable choice of ε, the first observation of a black raven reduces the posterior belief in H, confirming the theoretical analysis.

The main theorems extend this phenomenon beyond a single time step. Theorem 7 and Corollary 12 prove that, for the unnormalized Solomonoff prior, there exist time steps at which observing a black raven disconfirms H. Theorem 8 and Corollary 13 go further: for any computable infinite sequence consistent with H, the posterior under M decreases infinitely often. In contrast, Theorem 11 shows that when using the normalized prior M_norm, the number of such disconfirming steps is finite for any computable sequence consistent with H. Thus normalization mitigates but does not eliminate the effect; the unnormalized prior exhibits an unbounded number of violations.

The paper argues that these results are independent of the choice of universal Turing machine, relying only on the universal dominance property of Solomonoff’s prior. Consequently, Solomonoff induction asymptotically converges to the correct hypothesis (by Blackwell–Dubins merging of opinions), yet it systematically violates Nicod’s criterion during the learning process. The authors conclude that the paradox of confirmation (Hempel’s paradox) is resolved not by rejecting Solomonoff induction but by rejecting Nicod’s criterion as an unreasonable principle. They advocate retaining Solomonoff induction as a mathematically sound model of inductive reasoning while abandoning the intuition that every positive instance of a universal claim necessarily confirms it.