Statistical inference in non-commutative probability

Inge Svein Helland

Anilesh Mohari

Miklós Rédei

Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non‐commutative case. Mathematical no‐go theorems are recalled then which show that, in general, the stability can not be preserved in non‐commutative context. Two possible interpretations of the impossibility of generalization of Bayesian statistical inference to the non‐commutative case are offered, none of which seems to be completely satisfying.

Jyotiranjan Beuria

Abstract Classical probability theory provides the standard foundation for rational inference, yet human probability judgments are known to systematically violate Kolmogorovian axioms through phenomena such as conjunction fallacies and question-order effects. Whether large language models (LLMs) exhibit structurally analogous departures from classical probability remains an important empirical question with broad implications for AI evaluation and cognitive science. Here, we report a controlled experimental investigation of the probabilistic structure of LLM responses under systematically varied contextual prompts, treating model outputs as behavioral observables of a high-dimensional inference system. Using paradigms drawn from cognitive decision research, we evaluated multiple LLMs across tasks, including ambiguity judgments, belief revision, order effects, conjunction fallacies, and base-rate reasoning. Our experiments reveal systematic violations of the law of total probability accompanied by pronounced order-dependent effects across all models. Critically, these deviations are not random: they exhibit a structured interference form characterized by a phase-dependent term and a non-trivial upper bound, and the observed scaling relation demonstrates that the resulting probability assignments cannot be embedded within any single Kolmogorov probability space. These findings are consistent with a constrained non-commutative probabilistic structure, formally analogous to quantum probability models previously proposed for human cognition, without implying that neural networks implement quantum-physical processes. Our results establish non-commutative probability as a principled descriptive framework for contextual inference in large-scale artificial systems, and highlight the need for structural, beyond accuracy-based evaluation of probabilistic reasoning in modern AI.

Dror Bar-Natan

CHRISTOF WETTERICH

Non-commutative quantum physics at the atom scale can arise from coarse graining of a classical statistical ensemble at the Planck scale. Position and momentum of an isolated particle are classical observables which remain computable in terms of the coarse grained information. However, the commuting classical product of position and momentum observables is no longer defined in the coarse grained system, which is therefore described by incomplete statistics. The microphysical classical statistical ensemble at the Planck scale admits an alternative non-commuting product structure for position and momentum observables which is compatible with the coarse graining. Measurement correlations for isolated atoms are based on this non-commutative product structure. We present an explicit example for these ideas. It also realizes the discreteness of the spin observable within a microphysical classical statistical ensemble.