A Formal Correspondence Between Bayesian Inference Problems and the Heisenberg Representation

The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems. The likelihood function is first expressed in a Hilbert space through a quadratic norm of the difference between noisy observations and the response of a forward model depending on the unknown parameters ν, weighted by the inverse of the observational noise covariance. Based on the common mathematical structure of this quadratic form and an operator expectation value, the inverse observational noise covariance operator is identified, as a modeling hypothesis, with the initial observable A^(s,0)≡Γnoise−1. To establish this correspondence, a stochastic field with Matérn covariance C is considered and represented through the Karhunen–Loève expansion. The eigenvalues and eigenfunctions of this expansion are then used to define the state |Ψ⟩ within the proposed formalism and to construct the Hamiltonian governing the Heisenberg evolution. The main theorem formalizes the correspondence between the Bayesian likelihood and the initial Heisenberg observable under this modeling hypothesis. Limiting cases of the Matérn covariance are analyzed as ℓ→0 and ℓ→∞. Finally, numerical examples illustrate a non-trivial Heisenberg evolution, verify the invariance of the observable trace, and demonstrate the numerical equivalence between the Bayesian likelihood and the Heisenberg representation in the Karhunen–Loève basis, including standard and hierarchical Bayesian benchmarks.