When can non‐commutative statistical inference be Bayesian?

Abstract

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.