1,186,581 research outputs found
Independent natural extension for sets of desirable gambles
We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal sets of gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence
Independent Natural Extension for Infinite Spaces
We define and study the independent natural extension of two local
uncertainty models for the general case of infinite spaces, using the
frameworks of sets of desirable gambles and conditional lower previsions. In
contrast to Miranda and Zaffalon (2015), we adopt Williams-coherence instead of
Walley-coherence. We show that our notion of independent natural extension
always exists - whereas theirs does not - and that it satisfies various
convenient properties, including factorisation and external additivity. The
strength of these properties depends on the specific type of epistemic
independence that is adopted. In particular, epistemic event-independence is
shown to outperform epistemic atom-independence. Finally, the cases of lower
expectations, expectations, lower probabilities and probabilities are obtained
as special instances of our general definition. By applying our results to
these instances, we demonstrate that epistemic independence is indeed
epistemic, and that it includes the conventional notion of independence as a
special case.Comment: Parts of this contribution already appeared in an earlier conference
paper, the arXiv version of which is arXiv:1701.07295. The current version
extends this previous work, adding various new results and examples, and has
been submitted for publication in the ISIPTA 2017 special edition of the
International Journal of Approximate Reasonin
Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations
The concept of system signature was introduced by Samaniego for systems whose
components have i.i.d. lifetimes. We consider its extension to the continuous
dependent case and give an explicit expression for this extension as a
difference of weighted means of the structure function values. We then derive a
formula for the computation of the coefficients of these weighted means in the
special case of independent continuous lifetimes. Finally, we interpret this
extended concept of signature through a natural least squares approximation
problem
A law of the iterated logarithm sublinear expectations
In this paper, motivated by the notion of independent identically distributed
(IID) random variables under sub-linear expectations initiated by Peng, we
investigate a law of the iterated logarithm for capacities. It turns out that
our theorem is a natural extension of the Kolmogorov and the Hartman-Wintner
laws of the iterated logarithm
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