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This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
Haavelmo (1944) proposed a probabilistic structure for econometric modeling, aiming to make econometrics useful for decision making. His fundamental contribution has become thoroughly embedded in subsequent econometric research, yet it…
We present a general formalism with the aim of describing the situation of an entity, how it is, how it reacts to experiments, how we can make statistics with it, and how it changes under the influence of the rest of the universe. Therefore…
We celebrate this year hundred years of quantum mechanics but there is still no consensus regarding its interpretation and limitations. In this article we advocate the statistical contextual interpretation which is free of paradoxes. State…
We argue here that, as it happens in Classical and Quantum Mechanics, where it has been proven that alternative Hamiltonian descriptions can be compatible with a given set of equations of motion, the same holds true in the realm of…
In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear…
It is shown that the statistical conception of quantum mechanics is dynamical but not probabilistic, i.e. the statistical description in quantum mechanics is founded on dynamics. A use of the probability theory, when it takes place, is…
In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint…
R. Duncan Luce once mentioned in a conversation that he did not consider Kolmogorov's probability theory well-constructed because it treats stochastic independence as a "numerical accident," while it should be treated as a fundamental…
Beginning with a simple semantics for propositions, based on counting observations, it is shown that probabilistic and fuzzy logic correspond to two different heuristic assumptions regarding the combination of propositions whose evidence…
Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant…
This is a writeup of lectures on "statistics" that have evolved from the initial version for the 2009 Hadron Collider Physics Summer School at CERN to versions for other venues and, most recently, for the African School of Fundamental…
We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum…
Bohmian mechanics represents the universe as a set of paths with a probability measure defined on it. The way in which a mathematical model of this kind can explain the observed phenomena of the universe is examined in general. It is shown…
Everett's Relative State Interpretation has gained increasing interest due to the progress of understanding the role of decoherence. In order to fulfill its promise as a realistic description of the physical world, two postulates are…
Physics seeks to uncover the laws of Nature and express them through mathematical equations. Despite the vast diversity of natural phenomena, physical equations exhibit structural regularities that set them apart from arbitrary mathematical…
In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space…
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic…
Based on the concept of ensemble, it is proved in the manuscript that the probability amplitude function can also been used to describe the classical statistical system. The motion equations of probability amplitude functions of classical…
Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the…