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We describe the interface between measure theoretic probability and causal inference by constructing causal models on probability spaces within the potential outcomes framework. We find that measure theory provides a precise and instructive…
This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued…
Given a sequence of Borel probability measures on a Hausdorff space which satisfy a large deviation principle, we consider the corresponding sequence of measures formed by conditioning on a set $B$. If the large deviation rate function $I$…
Gravity-induced quantum interference is a remarkable effect that has already been confirmed experimentally, and it is a phenomenon in which quantum mechanics and gravity play simultaneously an important role. Additionally, a generalized…
An approach is proposed for inferring Granger causality between jointly stationary, Gaussian signals from quantized data. First, a necessary and sufficient rank criterion for the equality of two conditional Gaussian distributions is proved.…
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just…
A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lueders - von Neumann quantum…
The possibility to recover the which-way information, for example in the two slit experiment, is based on a natural but implicit assumption about the position of a particle {\it before} a position measurement is performed on it. This…
This paper explores the connection between a generalized Riesz electric energy and norms on the set of probability measures defined in terms of duality. We derive functional inequalities linking these two notions, recovering and…
According to Born's rule quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. This means that both contribute to any observed nonclassical effect that is usually…
We prove a Donsker and a Glivenko--Cantelli theorem for sequences of random discrete measures generalizing empirical measures. Those two results hold under standard conditions upon bracketing numbers of the indexing class of functions. As a…
The presence of latent variables can greatly complicate inferences about causal relations between measured variables from statistical data. In many cases, the presence of latent variables makes it impossible to determine for two measured…
In this paper, closed-form expressions are presented for the variation of the expectation of a given function due to changes in the probability measure used for the expectation. They unveil interesting connections with Gibbs probability…
Left on its own, a quantum state evolves deterministically under the Schr\"odinger Equation, forming superpositions. Upon measurement, however, a stochastic process governed by the Born rule collapses it to a single outcome. This dual…
The Born rule describes the probability of obtaining an outcome when measuring an observable of a quantum system. As it can only be tested by measuring many copies of the system under consideration, it does not hold for non-replicable…
In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time…
We overcome one of Bell's objections to `quantum measurement' by generalizing the definition to include systems outside the laboratory. According to this definition a {\sl generalized quantum measurement} takes place when the value of a…
The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only…
Suppose that particle detectors are placed along a Cauchy surface $\Sigma$ in Minkowski space-time, and consider a quantum theory with fixed or variable number of particles (i.e., using Fock space or a subspace thereof). It is…
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original…