Inductive Inference

Inductive inference is the computational and logical study of learning from data — the process of constructing general hypotheses from finite observations. Unlike deductive reasoning, which guarantees truth preservation, inductive inference operates under uncertainty: it generalizes beyond the observed cases, knowing that any generalization might be falsified by future data. The field asks not whether induction is justified — Hume's problem — but what can be inferred, by what algorithms, and with what guarantees.

The modern computational theory of inductive inference was developed by E. Mark Gold and later refined through the lens of Kolmogorov complexity and algorithmic randomness. Gold's framework distinguishes between identification in the limit — a learner that eventually converges to the correct hypothesis, though it never knows when it has converged — and finite identification — learning with explicit bounds on the number of examples required. These distinctions reveal that induction is not a single activity but a spectrum of learning tasks, each with different computational demands and different epistemic statuses.

The connection to Bayesian inference is deep but asymmetric. Bayesian updating provides a coherent framework for revising beliefs, but it requires a prior probability distribution over hypotheses — and the choice of prior is itself an inductive commitment. Algorithmic approaches to inductive inference, including minimum description length and Solomonoff induction, replace the arbitrary prior with a universal prior based on Kolmogorov complexity. The result is an objective but uncomputable inductive method: it defines the optimal learner, but no algorithm can implement it exactly.

The persistent philosophical suspicion of induction — the worry that it lacks deductive justification — is a category error masquerading as a deep problem. Induction does not need deductive justification; it needs a theory of what can be learned, from what data, by what computational resources. That theory exists, and it reveals that induction is not a philosophical mystery but a computational trade-off. The real question is not 'is induction valid?' but 'what is the price of learning, and who can afford it?' The answer depends on the structure of the hypothesis space, the regularity of the data source, and the computational budget of the learner — none of which are philosophical primitives, and all of which are systems-theoretic variables.