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Register a new userQuantum Probabilities as Behavioral Probabilities
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We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data.
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The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.
As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a ``naive space, which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR (coarsening at random) in the statistical literature characterizes when ``naive conditioning in a naive space works. We show that the CAR condition holds rather infrequently. We then consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, but show that there are no such conditions for MRE. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.
We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.
We use a stochastic birth-death model for a population of cells to estimate the normal tissue complication probability (NTCP) under a particular radiotherapy protocol. We specifically allow for interaction between cells, via a nonlinear logistic growth model. To capture some of the effects of intrinsic noise in the population we develop several approximations of NTCP, using Kramers-Moyal expansion techniques. These approaches provide an approximation to the first and second moments of a general first-passage time problem in the limit of large, but finite populations. We use this method to study NTCP in a simple model of normal cells and in a model of normal and damaged cells. We also study a combined model of normal tissue cells and tumour cells. Based on existing methods to calculate tumour control probabilities, and our procedure to approximate NTCP, we estimate the probability of complication free tumour control.
In the paper Relating Strong Behavioral Equivalences for Processes with Nondeterminism and Probabilities to appear in TCS, we present a comparison of behavioral equivalences for nondeterministic and probabilistic processes. In particular, we consider strong trace, failure, testing, and bisimulation equivalences. For each of these groups of equivalences, we examine the discriminating power of three variants stemming from three approaches that differ for the way probabilities of events are compared when nondeterministic choices are resolved via deterministic schedulers. The established relationships are summarized in a so-called spectrum. However, the equivalences we consider in that paper are only a small subset of those considered in the original spectrum of equivalences for nondeterministic systems introduced by Rob van Glabbeek. In this companion paper we we enlarge the spectrum by considering variants of trace equivalences (completed-trace equivalences), additional decorated-trace equivalences (failure-trace, readiness, and ready-trace equivalences), and variants of bisimulation equivalences (kernels of simulation, completed-simulation, failure-simulation, and ready-simulation preorders). Moreover, we study how the spectrum changes when randomized schedulers are used instead of deterministic ones.
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