Latent Coherence: Do Frontier Language Models' Probability Judgments Obey the Law of Total Probability?

arXiv:2610.31833v1 Announce Type: new Abstract: LawZero's Scientist AI trains a Predictor toward the Bayesian posterior, with an objective that penalizes inconsistent probabilities among latent explanations. We ask whether current frontier language models, with no such training, are already consistent in this sense zero-shot: do their probability judgments satisfy the law of total probability, $P(X)=\sum_z P(z)P(X|z)$, when the latent $z$ must be inferred from records in the context? In separate single-turn calls, models are asked for $P(X)$, for $P(z)$ over the latent's states, and for $P(X|z)$; a causal family applies the same test to $P(y|do(x))$ and the back-door adjustment. Every query has an exact ground-truth probability, and a subset of items is asked in three wordings to separate persistent incoherence from rewording noise. Across seven models, GPT-6.1 Sol and Claude Opus 5.5 are the most coherent (mean absolute residual 0.08 and 0.12 log-odds, about 2 percentage points). Claude Haiku 5.5, Gemini 3.8 Flash and DeepSeek V4-Pro show persistent incoherence (0.27-0.35 log-odds) that is mostly item-specific rather than a shared offset. Within every model, larger residuals go with larger errors against the truth (Spearman 0.40-0.62), so the residual, which needs no ground truth, flags unreliable answers. Reasoning length does not predict coherence, across models or within a model across rewordings of the same question. Simple consistency checks pass almost everywhere; the failures are specific to reasoning through a latent variable. Code and data: https://github.com/ppotash/latent-coherence