Integral Probability Metrics and Their Generating Classes of Functions

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

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Stochastic Orders Generated by Integrals: a Unified Study

Advances in Applied Probability ◽

10.1017/s0001867800028068 ◽

1997 ◽

Vol 29 (02) ◽

pp. 414-428 ◽

Cited By ~ 8

Author(s):

Alfred Müller

Keyword(s):

Functional Analysis ◽

Weak Convergence ◽

Likelihood Ratio ◽

Stochastic Orders ◽

Probability Measures ◽

Likelihood Ratio Order ◽

Unified Study ◽

Maximal Cone ◽

Class Of Functions

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

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Stochastic Orders Generated by Integrals: a Unified Study

Advances in Applied Probability ◽

10.2307/1428010 ◽

1997 ◽

Vol 29 (2) ◽

pp. 414-428 ◽

Cited By ~ 57

Author(s):

Alfred Müller

Keyword(s):

Functional Analysis ◽

Weak Convergence ◽

Likelihood Ratio ◽

Stochastic Orders ◽

Probability Measures ◽

Likelihood Ratio Order ◽

Unified Study ◽

Maximal Cone ◽

Class Of Functions

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

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Generalization Bounds with Integral Probability Metrics

Advances in Domain Adaption Theory ◽

10.1016/b978-1-78548-236-6.50005-2 ◽

2019 ◽

pp. 75-92

Author(s):

Ievgen Redko ◽

Amaury Habrard ◽

Emilie Morvant ◽

Marc Sebban ◽

Younès Bennani

Keyword(s):

Probability Metrics ◽

Generalization Bounds ◽

Integral Probability

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Non-parametric estimation of integral probability metrics

2010 IEEE International Symposium on Information Theory ◽

10.1109/isit.2010.5513626 ◽

2010 ◽

Cited By ~ 4

Author(s):

Bharath K. Sriperumbudur ◽

Kenji Fukumizu ◽

Arthur Gretton ◽

Bernhard Scholkopf ◽

Gert R. G. Lanckriet

Keyword(s):

Parametric Estimation ◽

Probability Metrics ◽

Non Parametric Estimation ◽

Integral Probability ◽

Non Parametric

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Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs

SSRN Electronic Journal ◽

10.2139/ssrn.3714012 ◽

2020 ◽

Author(s):

Tengyuan Liang

Keyword(s):

Probability Metrics ◽

Integral Probability

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A Unifying Framework for Probabilistic Validation Metrics

Journal of Verification Validation and Uncertainty Quantification ◽

10.1115/1.4045296 ◽

2019 ◽

Vol 4 (3) ◽

Author(s):

Paul Gardner ◽

Charles Lord ◽

Robert J. Barthorpe

Keyword(s):

Observational Data ◽

User Experience ◽

Probability Metrics ◽

Engineering Applications ◽

Validation Metrics ◽

Modeling Methods ◽

Small Set ◽

Probability Mass ◽

Integral Probability ◽

Quantities Of Interest

Abstract Probabilistic modeling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution for output quantities of interest. A challenge in applying probabilistic computer models (simulators) is validating output distributions against samples from observational data. An ideal validation metric is one that intuitively provides information on key differences between the simulator output and observational distributions, such as statistical distances/divergences. Within the literature, only a small set of statistical distances/divergences have been utilized for this task; often selected based on user experience and without reference to the wider variety available. As a result, this paper offers a unifying framework of statistical distances/divergences, categorizing those implemented within the literature, providing a greater understanding of their benefits, and offering new potential measures as validation metrics. In this paper, two families of measures for quantifying differences between distributions, that encompass the existing statistical distances/divergences within the literature, are analyzed: f-divergence and integral probability metrics (IPMs). Specific measures from these families are highlighted, providing an assessment of current and new validation metrics, with a discussion of their merits in determining simulator adequacy, offering validation metrics with greater sensitivity in quantifying differences across the range of probability mass.

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An Evaluation of Validation Metrics for Probabilistic Model Outputs

ASME 2018 Verification and Validation Symposium ◽

10.1115/vvs2018-9327 ◽

2018 ◽

Cited By ~ 2

Author(s):

Paul Gardner ◽

Charles Lord ◽

Robert J. Barthorpe

Keyword(s):

Statistical Method ◽

Probabilistic Model ◽

Probabilistic Models ◽

Statistical Moments ◽

Probabilistic Modelling ◽

Probability Metrics ◽

Engineering Applications ◽

Validation Metrics ◽

Modelling Methods ◽

Integral Probability

Probabilistic modelling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution, or summary statistical moments, for output quantities. A challenge in applying probabilistic models is validating output distributions. An ideal validation metric is one that intuitively provides information on key divergences between the output and validation distributions. Furthermore, it should be interpretable across different problems in order to informatively select the appropriate statistical method. In this paper, two families of measures for quantifying differences between distributions are compared: f-divergence and integral probability metrics (IPMs). Discussions and evaluation of these measures as validation metrics are performed with comments on ease of computation, interpretability and quantity of information provided.

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Proximity of Bachelier and Samuelson Models for Different Metrics

Review of Business and Economics Studies ◽

10.26794/2308-944x-2021-9-3-52-76 ◽

2021 ◽

Vol 9 (3) ◽

pp. 52-76

Author(s):

S. Smirnov ◽

D. Sotnikov

Keyword(s):

Option Price ◽

Model Parameters ◽

European Options ◽

Probability Metrics ◽

Lipschitz Continuous ◽

Oil Market ◽

Order Of Magnitude ◽

Kolmogorov Metric ◽

The Subject ◽

Probabilistic Metrics

This paper proposes a method of comparing the prices of European options, based on the use of probabilistic metrics, with respect to two models of price dynamics: Bachelier and Samuelson. In contrast to other studies on the subject, we consider two classes of options: European options with a Lipschitz continuous payout function and European options with a bounded payout function. For these classes, the following suitable probability metrics are chosen: the Fortet-Maurier metric, the total variation metric, and the Kolmogorov metric. It is proved that their computation can be reduced to computation of the Lambert in case of the Fortet-Mourier metric, and to the solution of a nonlinear equation in other cases. A statistical estimation of the model parameters in the modern oil market gives the order of magnitude of the error, including the magnitude of sensitivity of the option price, to the change in the volatility.

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