4,071 research outputs found
A Recursive Algorithm for Computing Inferences in Imprecise Markov Chains
We present an algorithm that can efficiently compute a broad class of
inferences for discrete-time imprecise Markov chains, a generalised type of
Markov chains that allows one to take into account partially specified
probabilities and other types of model uncertainty. The class of inferences
that we consider contains, as special cases, tight lower and upper bounds on
expected hitting times, on hitting probabilities and on expectations of
functions that are a sum or product of simpler ones. Our algorithm exploits the
specific structure that is inherent in all these inferences: they admit a
general recursive decomposition. This allows us to achieve a computational
complexity that scales linearly in the number of time points on which the
inference depends, instead of the exponential scaling that is typical for a
naive approach
Preference fusion and Condorcet's Paradox under uncertainty
Facing an unknown situation, a person may not be able to firmly elicit
his/her preferences over different alternatives, so he/she tends to express
uncertain preferences. Given a community of different persons expressing their
preferences over certain alternatives under uncertainty, to get a collective
representative opinion of the whole community, a preference fusion process is
required. The aim of this work is to propose a preference fusion method that
copes with uncertainty and escape from the Condorcet paradox. To model
preferences under uncertainty, we propose to develop a model of preferences
based on belief function theory that accurately describes and captures the
uncertainty associated with individual or collective preferences. This work
improves and extends the previous results. This work improves and extends the
contribution presented in a previous work. The benefits of our contribution are
twofold. On the one hand, we propose a qualitative and expressive preference
modeling strategy based on belief-function theory which scales better with the
number of sources. On the other hand, we propose an incremental distance-based
algorithm (using Jousselme distance) for the construction of the collective
preference order to avoid the Condorcet Paradox.Comment: International Conference on Information Fusion, Jul 2017, Xi'an,
Chin
Imprecise Markov chains and their limit behaviour
When the initial and transition probabilities of a finite Markov chain in
discrete time are not well known, we should perform a sensitivity analysis.
This can be done by considering as basic uncertainty models the so-called
credal sets that these probabilities are known or believed to belong to, and by
allowing the probabilities to vary over such sets. This leads to the definition
of an imprecise Markov chain. We show that the time evolution of such a system
can be studied very efficiently using so-called lower and upper expectations,
which are equivalent mathematical representations of credal sets. We also study
how the inferred credal set about the state at time n evolves as n goes to
infinity: under quite unrestrictive conditions, it converges to a uniquely
invariant credal set, regardless of the credal set given for the initial state.
This leads to a non-trivial generalisation of the classical Perron-Frobenius
Theorem to imprecise Markov chains.Comment: v1: 28 pages, 8 figures; v2: 31 pages, 9 figures, major revision
after review: added, modified, and removed material (no results dropped,
results added), moved proofs to an appendi
Probability-free pricing of adjusted American lookbacks
Consider an American option that pays G(X^*_t) when exercised at time t,
where G is a positive increasing function, X^*_t := \sup_{s\le t}X_s, and X_s
is the price of the underlying security at time s. Assuming zero interest
rates, we show that the seller of this option can hedge his position by trading
in the underlying security if he begins with initial capital
X_0\int_{X_0}^{\infty}G(x)x^{-2}dx (and this is the smallest initial capital
that allows him to hedge his position). This leads to strategies for trading
that are always competitive both with a given strategy's current performance
and, to a somewhat lesser degree, with its best performance so far. It also
leads to methods of statistical testing that avoid sacrificing too much of the
maximum statistical significance that they achieve in the course of
accumulating data.Comment: 28 pages, 1 figur
Computable randomness is about more than probabilities
We introduce a notion of computable randomness for infinite sequences that
generalises the classical version in two important ways. First, our definition
of computable randomness is associated with imprecise probability models, in
the sense that we consider lower expectations (or sets of probabilities)
instead of classical 'precise' probabilities. Secondly, instead of binary
sequences, we consider sequences whose elements take values in some finite
sample space. Interestingly, we find that every sequence is computably random
with respect to at least one lower expectation, and that lower expectations
that are more informative have fewer computably random sequences. This leads to
the intriguing question whether every sequence is computably random with
respect to a unique most informative lower expectation. We study this question
in some detail and provide a partial answer
Accept & Reject Statement-Based Uncertainty Models
We develop a framework for modelling and reasoning with uncertainty based on
accept and reject statements about gambles. It generalises the frameworks found
in the literature based on statements of acceptability, desirability, or
favourability and clarifies their relative position. Next to the
statement-based formulation, we also provide a translation in terms of
preference relations, discuss---as a bridge to existing frameworks---a number
of simplified variants, and show the relationship with prevision-based
uncertainty models. We furthermore provide an application to modelling symmetry
judgements.Comment: 35 pages, 17 figure
Continuity of the shafer-Vovk-Ville operator
Kolmogorovâs axiomatic framework is the best-known approach to describing probabilities and, due to its use of the Lebesgue integral, leads to remarkably strong continuity properties. However, it relies on the specification of a probability measure on all measurable events. The game-theoretic framework proposed by Shafer and Vovk does without this restriction. They define global upper expectation operators using local betting options. We study the continuity properties of these more general operators. We prove that they are continuous with respect to upward convergence and show that this is not the case for downward convergence. We also prove a version of Fatouâs Lemma in this more general context. Finally, we prove their continuity with respect to point-wise limits of two-sided cuts
Lung Transplantation in the United States, 1999â2008
This article highlights trends and changes in lung and heartâlung transplantation in the United States from 1999 to 2008. While adult lung transplantation grew significantly over the past decade, rates of heartâlung and pediatric lung transplantation have remained low. Since implementation of the lung allocation score (LAS) donor allocation system in 2005, decreases in the number of active waiting list patients, waiting times for lung transplantation and death rates on the waiting list have occurred. However, characteristics of recipients transplanted in the LAS era differed from those transplanted earlier. The proportion of candidates undergoing lung transplantation for chronic obstructive pulmonary disease decreased, while increasing for those with pulmonary fibrosis. In the LAS era, older, sicker and previously transplanted candidates underwent transplantation more frequently compared with the previous era. Despite these changes, when compared with the pre-LAS era, 1-year survival after lung transplantation did not significantly change after LAS inception. The long-term effects of the change in the characteristics of lung transplant recipients on overall outcomes for lung transplantation remain unknown. Continued surveillance and refinements to the LAS system will affect the distribution and types of candidates transplanted and hopefully lead to improved system efficiency and outcomes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/79080/1/j.1600-6143.2010.03055.x.pd
Completely monotone outer approximations of lower probabilities on ïŹnite possibility spaces
Drawing inferences from general lower probabilities on finite possibility spaces usually involves solving linear programming problems. For some applications this may be too computationally demanding. Some special classes of lower probabilities allow for using computationally less demanding techniques. One such class is formed by the completely monotone lower probabilities, for which inferences can be drawn efficiently once their Möbius transform has been calculated. One option is therefore to draw approximate inferences by using a completely monotone approximation to a general lower probability; this must be an outer approximation to avoid drawing inferences that are not implied by the approximated lower probability. In this paper, we discuss existing and new algorithms for performing this approximation, discuss their relative strengths and weaknesses, and illustrate how each one works and performs
- âŠ