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Optimal uncertainty quantification for legacy data observations of Lipschitz functions

T. J. Sullivan, M. McKerns, D. Meyer, F. Theil, H. Owhadi, M. Ortiz (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter...

Optimality conditions for maximizers of the information divergence from an exponential family

František Matúš (2007)

Kybernetika

The information divergence of a probability measure P from an exponential family ℰ over a finite set is defined as infimum of the divergences of P from Q subject to Q ∈ ℰ . All directional derivatives of the divergence from ℰ are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for P to be a maximizer of the divergence from ℰ are presented, including new ones when P  is not projectable to ℰ .

Optimality of the least weighted squares estimator

Libor Mašíček (2004)

Kybernetika

The present paper deals with least weighted squares estimator which is a robust estimator and it generalizes classical least trimmed squares. We will prove n -consistency and asymptotic normality for any sequence of roots of normal equation for location model. The influence function for general case is calculated. Finally optimality of this estimator is discussed and formula for most B-robust and most V-robust weights is derived.

Optimally approximating exponential families

Johannes Rauh (2013)

Kybernetika

This article studies exponential families ℰ on finite sets such that the information divergence D ( P ∥ ℰ ) of an arbitrary probability distribution from ℰ is bounded by some constant D > 0 . A particular class of low-dimensional exponential families that have low values of D can be obtained from partitions of the state space. The main results concern optimality properties of these partition exponential families. The case where D = log ( 2 ) is studied in detail. This case is special, because if D < log ( 2 ) , then ℰ contains all probability...

Optimisation in space of measures and optimal design

Ilya Molchanov, Sergei Zuyev (2004)

ESAIM: Probability and Statistics

The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes...

Optimisation in space of measures and optimal design

Ilya Molchanov, Sergei Zuyev (2010)

ESAIM: Probability and Statistics

The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework...

Optimization of parameters in the Menzerath–Altmann law

Ján Andres, Lubomír Kubáček, Jitka Machalová, Michaela Tučková (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Four formulas of the Menzerath–Altmann law are tested from the point of view of their applicability and suitability. The accuracy of related approximations of measured data is examined by the least square method at first. Then the accuracy of calculated parameters in the formulas under consideration is compared statistically. The influence of neglecting parameter c is investigated as well. Finally, the obtained results are discussed by means of an illustrative example from quantitative linguistics....

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