Conditional expectations of quantum observables I: Bohm momentum as best predictor of momentum given position

Given the state of a quantum system, we define the conditional expectation of an observable with respect to a second observable, or commuting family of observables, as the real-valued function of the latter which best approximates the observable, in the sense of minimizing the quantum mechanical expectation of the difference squared. This corresponds to the notion of best predictor of a random variable by another random variable in non-parametric statistical regression. A basic example is the conditional expectation of the quantum mechanical momentum with respect to position, which can be identified with the Bohm momentum. We re-examine the de Broglie-Bohm theory from this perspective. As a second example we consider the simplest quantum system, that of a spin-$1/2 $ paticle in a magnetic field. We study the conditional expectations of the Pauli spin-operators S₁, S₂, S₃ with respect to S₃, and determine their dynamics if the pure-state wave-function solves the relevant Schr"odinger equation. Considering S₃ as a beable we can, following Bell, introduce a generalized guidance law in the form of a Markov jump process on the spectrum of S₃ which is compatible with the time evolution of the Born probabilities of the two eigenstates of S₃ . The conditional expectations of the remaining spin operators are then shown to follow a piece-wise deterministic Markov process, defined by an explicit SDE. We finally examine the relationship between the conditional expectations of this paper, and the von Neumann-algebraic conditional expectation onto the sub-algebra generated by the commuting family B when the joint spectrum of the latter is discrete, and derive a general expression for the conditional expectation dynamics in that context.