18 research outputs found
Envelopes of conditional probabilities extending a strategy and a prior probability
Any strategy and prior probability together are a coherent conditional
probability that can be extended, generally not in a unique way, to a full
conditional probability. The corresponding class of extensions is studied and a
closed form expression for its envelopes is provided. Then a topological
characterization of the subclasses of extensions satisfying the further
properties of full disintegrability and full strong conglomerability is given
and their envelopes are studied.Comment: 2
Numerical Methods for Infinite Decision-Making Processes
The new computational methodology due to Yaroslav Sergeyev (see [25–27]) makes it possible to evaluate numerically the terminal features of complete, sequential decision-making processes. By standard numerical methods, these processes have indeterminate features or seem to support paradoxical conclusions. We show that they are better regarded as a class of problems for which the numerical methods based on Sergeyev’s methodology provide a uniform technique of resolution
Additivity Requirements in Classical and Quantum Probability
The discussion of different principles of additivity (finite vs. countable vs. complete additivity) for probability functions has been largely focused on the personalist interpretation of probability. Very little attention has been given to additivity principles for physical probabilities. The form of additivity for quantum probabilities is determined by the algebra of observables that characterize a physical system and the type of quantum state that is realizable and preparable for that system. We assess arguments designed to show that only normal quantum states are realizable and preparable and, therefore, quantum probabilities satisfy the principle of complete additivity. We underscore the little remarked fact that unless the dimension of the Hilbert space is incredibly large, complete additivity in ordinary non-relativistic quantum mechanics (but not in relativistic quantum field theory) reduces to countable additivity. We then turn to ways in which knowledge of quantum probabilities may constrain rational credence about quantum events and, thereby, constrain the additivity principle satisfied by rational credence functions
Additivity Requirements in Classical and Quantum Probability
The discussion of different principles of additivity (finite vs. countable vs. complete additivity) for probability functions has been largely focused on the personalist interpretation of probability. Very little attention has been given to additivity principles for physical probabilities. The form of additivity for quantum probabilities is determined by the algebra of observables that characterize a physical system and the type of quantum state that is realizable and preparable for that system. We assess arguments designed to show that only normal quantum states are realizable and preparable and, therefore, quantum probabilities satisfy the principle of complete additivity. We underscore the little remarked fact that unless the dimension of the Hilbert space is incredibly large, complete additivity in ordinary non-relativistic quantum mechanics (but not in relativistic quantum field theory) reduces to countable additivity. We then turn to ways in which knowledge of quantum probabilities may constrain rational credence about quantum events and, thereby, constrain the additivity principle satisfied by rational credence functions
2-coherent and 2-convex Conditional Lower Previsions
In this paper we explore relaxations of (Williams) coherent and convex
conditional previsions that form the families of -coherent and -convex
conditional previsions, at the varying of . We investigate which such
previsions are the most general one may reasonably consider, suggesting
(centered) -convex or, if positive homogeneity and conjugacy is needed,
-coherent lower previsions. Basic properties of these previsions are
studied. In particular, we prove that they satisfy the Generalized Bayes Rule
and always have a -convex or, respectively, -coherent natural extension.
The role of these extensions is analogous to that of the natural extension for
coherent lower previsions. On the contrary, -convex and -coherent
previsions with either are convex or coherent themselves or have no
extension of the same type on large enough sets. Among the uncertainty concepts
that can be modelled by -convexity, we discuss generalizations of capacities
and niveloids to a conditional framework and show that the well-known risk
measure Value-at-Risk only guarantees to be centered -convex. In the final
part, we determine the rationality requirements of -convexity and
-coherence from a desirability perspective, emphasising how they weaken
those of (Williams) coherence.Comment: This is the authors' version of a work that was accepted for
publication in the International Journal of Approximate Reasoning, vol. 77,
October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003,
http://www.sciencedirect.com/science/article/pii/S0888613X1630079