A formal correspondence between Bayesian inference problems and the Heisenberg representation

Abstract

The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems by identifying the observational noise operator with the initial observable. The initial observable is linked to the quadratic likelihood norm in Hilbert space, which depends on noisy observations and a direct model parameterized by unknown parameters. Using the Karhunen-Loeve expansion, the quantum state is expressed in terms of the eigenvalues and eigenfunctions of the Matern covariance. The likelihood probability is then expressed in terms of the observed state, and the model-dependent state, and is rewritten as a quadratic norm in terms of the quantum states, where the observational noise covariance is proposed to be identified with the initial observable. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the time invariance of the observable, and the equivalence between Heisenberg evolution and Bayesian posterior probability. Limiting cases of the Matern covariance are analyzed as the length parameter tends to zero and infinity. Finally, numerical results support the analytical findings.