1,246 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
A statistical inference method for the stochastic reachability analysis.
The main contribution of this paper is the characterization of reachability problem associated to stochastic hybrid systems in terms of imprecise probabilities. This provides the connection between reachability problem and Bayesian statistics. Using generalised Bayesian statistical inference, a new concept of conditional reach set probabilities is defined. Then possible algorithms to compute the reach set probabilities are derived
Discrete time models for bid-ask pricing under Dempster-Shafer uncertainty
As is well-known, real financial markets depart from simplifying hypotheses of classical no-arbitrage pricing theory. In particular, they show the presence of frictions in the form of bid-ask spread. For this reason, the aim of the thesis is to provide a model able to manage these situations, relying on a non-linear pricing rule defined as (discounted) Choquet integral with respect to a belief function. Under the partially resolving uncertainty principle, we generalize the first fundamental theorem of asset pricing in the context of belief functions. Furthermore, we show that a generalized arbitrage-free lower pricing rule can be characterized as a (discounted) Choquet expectation with respect to an equivalent inner approximating (one-step) Choquet martingale belief function. Then, we generalize the Choquet pricing rule dinamically: we characterize a reference belief function such that a multiplicative binomial process satisfies a suitable version of time-homogeneity and Markov properties and we derive the induced conditional Choquet expectation operator. In a multi-period market with a risky asset admitting bid-ask spread, we assume that its lower price process is modeled by the proposed time-homogeneous Markov multiplicative binomial process. Here, we generalize the theorem of change of measure, proving the existence of an equivalent one-step Choquet martingale belief function. Then, we prove that the (discounted) lower price process of a European derivative is a one-step Choquet martingale and a k-step Choquet super-martingale, for k ≥ 2
A Particular Upper Expectation as Global Belief Model for Discrete-Time Finite-State Uncertain Processes
To model discrete-time finite-state uncertain processes, we argue for the use
of a global belief model in the form of an upper expectation that is the most
conservative one under a set of basic axioms. Our motivation for these axioms,
which describe how local and global belief models should be related, is based
on two possible interpretations for an upper expectation: a behavioural one
similar to Walley's, and an interpretation in terms of upper envelopes of
linear expectations. We show that the most conservative upper expectation
satisfying our axioms, that is, our model of choice, coincides with a
particular version of the game-theoretic upper expectation introduced by Shafer
and Vovk. This has two important implications: it guarantees that there is a
unique most conservative global belief model satisfying our axioms; and it
shows that Shafer and Vovk's model can be given an axiomatic characterisation
and thereby provides an alternative motivation for adopting this model, even
outside their game-theoretic framework. Finally, we relate our model to the
upper expectation resulting from a traditional measure-theoretic approach. We
show that this measure-theoretic upper expectation also satisfies the proposed
axioms, which implies that it is dominated by our model or, equivalently, the
game-theoretic model. Moreover, if all local models are precise, all three
models coincide.Comment: Extension of the conference paper `In Search of a Global Belief Model
for Discrete-Time Uncertain Processes
A method of classification for multisource data in remote sensing based on interval-valued probabilities
An axiomatic approach to intervalued (IV) probabilities is presented, where the IV probability is defined by a pair of set-theoretic functions which satisfy some pre-specified axioms. On the basis of this approach representation of statistical evidence and combination of multiple bodies of evidence are emphasized. Although IV probabilities provide an innovative means for the representation and combination of evidential information, they make the decision process rather complicated. It entails more intelligent strategies for making decisions. The development of decision rules over IV probabilities is discussed from the viewpoint of statistical pattern recognition. The proposed method, so called evidential reasoning method, is applied to the ground-cover classification of a multisource data set consisting of Multispectral Scanner (MSS) data, Synthetic Aperture Radar (SAR) data, and digital terrain data such as elevation, slope, and aspect. By treating the data sources separately, the method is able to capture both parametric and nonparametric information and to combine them. Then the method is applied to two separate cases of classifying multiband data obtained by a single sensor. In each case a set of multiple sources is obtained by dividing the dimensionally huge data into smaller and more manageable pieces based on the global statistical correlation information. By a divide-and-combine process, the method is able to utilize more features than the conventional maximum likelihood method
Coherent frequentism
By representing the range of fair betting odds according to a pair of
confidence set estimators, dual probability measures on parameter space called
frequentist posteriors secure the coherence of subjective inference without any
prior distribution. The closure of the set of expected losses corresponding to
the dual frequentist posteriors constrains decisions without arbitrarily
forcing optimization under all circumstances. This decision theory reduces to
those that maximize expected utility when the pair of frequentist posteriors is
induced by an exact or approximate confidence set estimator or when an
automatic reduction rule is applied to the pair. In such cases, the resulting
frequentist posterior is coherent in the sense that, as a probability
distribution of the parameter of interest, it satisfies the axioms of the
decision-theoretic and logic-theoretic systems typically cited in support of
the Bayesian posterior. Unlike the p-value, the confidence level of an interval
hypothesis derived from such a measure is suitable as an estimator of the
indicator of hypothesis truth since it converges in sample-space probability to
1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly
extended to vector parameters of interest. The derivation of upper and lower
confidence levels from valid and nonconservative set estimators is formalize
A Dutch book coherence condition for conditional completely alternating Choquet expectations
Stemming from de Finetti’s coherence for finitely additive (conditional) probabilities, the paradigm of coherence has been extended to other uncertainty calculi. We study the notion of coherence for conditional completely alternating Choquet expectations, providing an avoiding Dutch book like condition
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