214,453 research outputs found
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
A desirability-based axiomatisation for coherent choice functions
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules to imprecise-probabilistic uncertainty models. We provide them with a clear interpretation in terms of attitudes towards gambling, borrowing ideas from the theory of sets of desirable gambles, and we use this interpretation to derive a set of basic axioms. We show that these axioms lead to a full-fledged theory of coherent choice functions, which includes a representation in terms of sets of desirable gambles, and a conservative inference method
A Desirability-Based Axiomatisation for Coherent Choice Functions
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that typically arise from applying decision
rules to imprecise-probabilistic uncertainty models. We provide them with a
clear interpretation in terms of attitudes towards gambling, borrowing ideas
from the theory of sets of desirable gambles, and we use this interpretation to
derive a set of basic axioms. We show that these axioms lead to a full-fledged
theory of coherent choice functions, which includes a representation in terms
of sets of desirable gambles, and a conservative inference method
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions
Coherent States for 3d Deformed Special Relativity: semi-classical points in a quantum flat spacetime
We analyse the quantum geometry of 3-dimensional deformed special relativity
(DSR) and the notion of spacetime points in such a context, identified with
coherent states that minimize the uncertainty relations among spacetime
coordinates operators. We construct this system of coherent states in both the
Riemannian and Lorentzian case, and study their properties and their geometric
interpretation.Comment: RevTeX4, 20 page
Believing Probabilistic Contents: On the Expressive Power and Coherence of Sets of Sets of Probabilities
Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Mossâ account of probabilistic knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower
probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch
bookability iff they have consistent probabilistic beliefs
An Information-Theoretic Measure of Uncertainty due to Quantum and Thermal Fluctuations
We study an information-theoretic measure of uncertainty for quantum systems.
It is the Shannon information of the phase space probability distribution
\la z | \rho | z \ra , where |z \ra are coherent states, and is the
density matrix. The uncertainty principle is expressed in this measure as . For a harmonic oscillator in a thermal state, coincides with von
Neumann entropy, - \Tr(\rho \ln \rho), in the high-temperature regime, but
unlike entropy, it is non-zero at zero temperature. It therefore supplies a
non-trivial measure of uncertainty due to both quantum and thermal
fluctuations. We study as a function of time for a class of non-equilibrium
quantum systems consisting of a distinguished system coupled to a heat bath. We
derive an evolution equation for . For the harmonic oscillator, in the
Fokker-Planck regime, we show that increases monotonically. For more
general Hamiltonians, settles down to monotonic increase in the long run,
but may suffer an initial decrease for certain initial states that undergo
``reassembly'' (the opposite of quantum spreading). Our main result is to
prove, for linear systems, that at each moment of time has a lower bound
, over all possible initial states. This bound is a generalization
of the uncertainty principle to include thermal fluctuations in non-equilibrium
systems, and represents the least amount of uncertainty the system must suffer
after evolution in the presence of an environment for time .Comment: 36 pages (revised uncorrupted version), Report IC 92-93/2
About Lorentz invariance in a discrete quantum setting
A common misconception is that Lorentz invariance is inconsistent with a
discrete spacetime structure and a minimal length: under Lorentz contraction, a
Planck length ruler would be seen as smaller by a boosted observer. We argue
that in the context of quantum gravity, the distance between two points becomes
an operator and show through a toy model, inspired by Loop Quantum Gravity,
that the notion of a quantum of geometry and of discrete spectra of geometric
operators, is not inconsistent with Lorentz invariance. The main feature of the
model is that a state of definite length for a given observer turns into a
superposition of eigenstates of the length operator when seen by a boosted
observer. More generally, we discuss the issue of actually measuring distances
taking into account the limitations imposed by quantum gravity considerations
and we analyze the notion of distance and the phenomenon of Lorentz contraction
in the framework of ``deformed (or doubly) special relativity'' (DSR), which
tentatively provides an effective description of quantum gravity around a flat
background. In order to do this we study the Hilbert space structure of DSR,
and study various quantum geometric operators acting on it and analyze their
spectral properties. We also discuss the notion of spacetime point in DSR in
terms of coherent states. We show how the way Lorentz invariance is preserved
in this context is analogous to that in the toy model.Comment: 25 pages, RevTe
The role of phase space geometry in Heisenberg's uncertainty relation
Aiming towards a geometric description of quantum theory, we study the
coherent states-induced metric on the phase space, which provides a geometric
formulation of the Heisenberg uncertainty relations (both the position-momentum
and the time-energy ones). The metric also distinguishes the original
uncertainty relations of Heisenberg from the ones that are obtained from
non-commutativity of operators. Conversely, the uncertainty relations can be
written in terms of this metric only, hence they can be formulated for any
physical system, including ones with non-trivial phase space. Moreover, the
metric is a key ingredient of the probability structure of continuous-time
histories on phase space. This fact allows a simple new proof the impossibility
of the physical manifestation of the quantum Zeno and anti-Zeno paradoxes.
Finally, we construct the coherent states for a spinless relativistic particle,
as a non-trivial example by which we demonstrate our results.Comment: 28 pages, late
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