Optimization vs. Enforcement: Invariant Drift in Probabilistic Systems

This paper analyzes invariant preservation in probabilistic inference systems, including large language models, through the lens of Bayesian updating and information geometry. We contrast monotonic deduction (where conclusions are preserved under added premises) with non-monotonic Bayesian updating (where new evidence can reduce posterior confidence). Sequential updates form a multiplicative process governed by the geometric mean of likelihood ratios. We show that when the expected log-likelihood ratio for invariant preservation is negative, posterior mass assigned to the invariant hypothesis decays exponentially with the number of updates. Under the assumption that update evidence is generated by a background “median generator” distribution Q that does not encode a domain-specific invariant, this expected log-likelihood ratio equals -DKL (Q || PH), where PH represents the invariant-preserving distribution. This identifies a precise sufficient condition for invariant drift in optimization-only systems absent explicit constraint enforcement.