DOCSLIB.ORG

Imprecise Probability Analysis for Integrated Assessment of Climate Change

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Imprecise Probability Analysis for Integrated Assessment of Climate Change Imprecise Probability Analysis for Integrated Assessment of Climate Change D I S S E R T A T I O N zur Erlangung des akademischen Grades \doctor rerum naturalium" (Dr. rer. nat.) in der Wissenschaftsdisziplin \Theoretische Physik" eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Potsdam vorgelegt von Elmar Kriegler Potsdam, im Februar 2005 When there is little information on which to base our conclusions, we cannot expect reasoning (no matter how clever or thorough) to reveal a most probable hypothesis or a uniquely reasonable course of action. There are limits to the power of reason. Peter Walley, 1991, in Statistical Reasoning with Imprecise Probabilities, London, pg. 2 In a predestinate world, decision would be illusory; in a world of perfect foreknowledge, empty; in a world without natural order, powerless. Our intuitive attitude to life implies non-illusory, non-empty, non-powerless decision... Since decision in this sense excludes both perfect foresight and anarchy in nature, it must be defined as choice in face of bounded uncertainty. George L.S. Shackle, 1961, in Decision, Order, and Time in Human Affairs, Cambridge, pg. 43 Abstract We present an application of imprecise probability theory to the quantification of uncertainty in the integrated assessment of climate change. Our work is motivated by the fact that uncertainty about climate change is pervasive, and therefore requires a thorough treatment in the integrated assessment process. Classical probability theory faces some severe difficulties in this respect, since it cannot capture very poor states of information in a satisfactory manner. A more general framework is provided by imprecise probability theory, which offers a similarly firm evidential and behavioural foundation, while at the same time allowing to capture more diverse states of information. An imprecise probability describes the information in terms of lower and upper bounds on probability. For the purpose of our imprecise probability analysis, we construct a diffusion ocean energy balance climate model that parameterises the global mean temperature response to secular trends in the radiative forcing in terms of climate sensitivity and effective vertical ocean heat diffusivity. We compare the model behaviour to the 20th century temperature record in order to derive a likelihood function for these two parameters and the forcing strength of anthropogenic sulphate aerosols. Results show a strong positive correlation between climate sensitivity and ocean heat diffusivity, and between climate sensitivity and absolute strength of the sulphate forcing. We have applied a series of statistical tests to the residual stochasticity, on the basis of which only values of climate sensitivity below 1:1 K, and a sulphate aerosol cooling above 2 1:75 W m− in the year 1990 could be rejected at the 5% significance level. We identify two suitable imprecise probability classes, probability boxes and "-contamination models, for an efficient representation of the uncertainty about the climate model parameters and provide an algorithm to combine the information content of these two classes into a single imprecise probability representation described by a belief function. We construct a belief func- tion for the prior parameter uncertainty from a set of probability estimates in the literature (climate sensitivity, sulphate aerosol cooling) and observational estimates of 20th century ocean heat uptake (ocean heat diffusivity). For the purpose of updating the prior with the likelihood function, we establish a methodological framework that allows us to perform the updating pro- cedure efficiently on the entire event space and for two different updating rules: Dempster's rule of conditioning and the Generalised Bayes' rule. Dempster's rule yields a posterior belief function in good qualitative agreement with previous studies that tried to constrain climate sensitivity and sulphate aerosol cooling. It allocates small weight to high values of climate sensitivity (upper 95% quantile: T 6:9 K) and excludes low values of climate sensitivity 2x ≈ T2x < 1:5 K with almost certainty. Moreover, the sulphate aerosol forcing in the year 1990 can 2 2 be constrained to [ 1:53 W m− ; 0:33 W m− ] with 99% lower confidence. In contrast, we are − − not able to produce meaningful imprecise posterior probability bounds from the application of the Generalised Bayes' Rule. We can attribute this result mainly to our choice of representing the prior uncertainty by a belief function. We project the Dempster-updated belief function for the climate model parameters onto es- timates of future global mean temperature change under several emissions scenarios for the 21st century, and several long-term stabilisation policies. The upper end of the warming estimates is dominated by the possibility of high values of climate sensitivity, and exceeds corresponding es- timates of the Intergovernmental Panel on Climate Change by more than 30%. At the low end, it is very unlikely that the warming in the 21st century will remain below 2 Kelvin in the absence of policy interventions. Within the limitations of our analysis we find that it requires a stringent stabilisation level of around 450 ppm CO2 equivalent to obtain a non-negligible lower proba- bility of limiting the warming to 2 Kelvin. We discuss several frameworks of decision-making iii under ambiguity and show that they can lead to a variety of, possibly imprecise, climate policy recommendations. We find, however, that poor states of information do not necessarily impede a useful policy advice. We conclude that imprecise probabilities constitute indeed a promising candidate for the adequate treatment of uncertainty in the integrated assessment of climate change. We have constructed prior belief functions that allow much weaker assumptions on the prior state of information than a prior probability would require and, nevertheless, can be propagated through the entire assessment process. As a caveat, the updating issue needs further investigation. Belief functions constitute only a sensible choice for the prior uncertainty representation if more restrictive updating rules than the Generalised Bayes' Rule are available. iv Zusammenfassung Diese Arbeit untersucht die Eignung der Theorie der unscharfen Wahrscheinlichkeiten fur¨ die Beschreibung der Unsicherheit in der integrierten Analyse des Klimawandels. Die wissenschaft- liche Unsicherheit bezuglic¨ h vieler Aspekte des Klimawandels ist betr¨achtlich, so dass ihre an- gemessene Beschreibung von großer Wichtigkeit ist. Die klassische Wahrscheinlichkeitstheorie weist in diesem Zusammenhang einige Probleme auf, da sie Zust¨ande sehr geringer Informa- tion nicht zufriedenstellend beschreiben kann. Die unscharfe Wahrscheinlichkeitstheorie bietet ein gleichermaßen fundiertes Theoriegeb¨aude, welches jedoch eine gr¨oßere Flexibilit¨at bei der Beschreibung verschiedenartiger Informationszust¨ande erlaubt. Unscharfe Wahrscheinlichkeiten erfassen solche Informationszust¨ande durch die Spezifizierung von unteren und oberen Grenzen an zul¨assige Werte der Wahrscheinlichkeit. Unsere Analyse des Klimawandels beruht auf einem Energiebilanzmodell mit diffusivem Ozean, welches die globale Temperaturantwort auf eine Anderung¨ der Strahlungsbilanz in Abh¨angigkeit von zwei Parametern beschreibt: die Klimasensitivit¨at, und die effektive vertikale W¨armediffusivit¨at im Ozean. Wir vergleichen das Modellverhalten mit den Temperaturmessun- gen des 20. Jahrhunderts, um eine sogenannte Likelihood-Funktion fur¨ die Hypothesen zu diesen beiden Parametern sowie dem kuhlenden¨ Einfluss der Sulfataerosole zu ermitteln. Im Ergebnis zeigt sich eine stark positive Korrelation zwischen Klimasensitivit¨at und W¨armediffusivit¨at im Ozean, und Klimasensitivit¨at und kuhlendem¨ Einfluss der Sulfataerosole. Wir haben eine Rei- he von statistischen Tests angewandt, um den Raum m¨oglicher Hypothesen einzuengen. Auf 2 diese Weise lassen sich lediglich eine Strahlungswirkung der Sulfataerosole, die 1:75 W m− − im Jahr 1990 unterschritten hat, und eine Klimasensitivit¨at unterhalb von 1:1 K mit einem Signifikanzniveau von 5% ausschließen. Fur¨ die effiziente Beschreibung der Parameterunsicherheit ziehen wir zwei geeignete Modell- typen aus der unscharfen Wahrscheinlichkeitstheorie heran: Wahrscheinlichkeitsverteilungsb¨an- der und sogenannte "-Kontaminationsmodelle. Wir formulieren einen Algorithmus, der den In- formationsgehalt beider Modelle vereint und durch eine sogenannte Belief-Funktion beschreibt. Mit Hilfe dieses Algorithmus konstruieren wir Belief-Funktionen fur¨ die A-priori-Parameter- unsicherheit auf der Grundlage von divergierenden Wahrscheinlichkeitssch¨atzungen in der Li- teratur (Klimasensitivit¨at, Strahlungswirkung des Sulfats) bzw. auf der Grundlage von auf Beobachtungsdaten beruhenden Sch¨atzungen der W¨armeaufnahme des Ozeans in der zweiten H¨alfte des 20. Jahrhunderts (W¨armediffusivit¨at im Ozean). Wir leiten eine Methode her, um die A-priori-Belief-Funktion im Lichte der Likelihood-Funktion zu aktualisieren. Dabei ziehen wir zwei verschiedene Regeln zur Durchfuhrung¨ des Lernprozesses in Betracht: die Dempster- sche Regel und die verallgemeinerte Bayessche Regel. Durch Anwendung der Dempsterschen Regel erhalten wir eine A-posteriori-Belief-Funktion, deren Informationsgehalt qualitativ mit den Ergebnissen bisheriger Studien ub¨ ereinstimmt, die eine Einschr¨ankung der Unsicherheit ub¨ er die Klimasensitivit¨at und die kuhlende¨ Wirkung der Sulfataerosole versucht haben. Ei-
Load more

Recommended publications
  • Imprecise Probability in Risk Analysis
    IMPRECISE PROBABILITY IN RISK ANALYSIS D. Dubois IRIT-CNRS, Université Paul Sabatier 31062 TOULOUSE FRANCE Outline 1. Variability vs incomplete information 2. Blending set-valued and probabilistic representations : uncertainty theories 3. Possibility theory in the landscape 4. A methodology for risk-informed decision- making 5. Some applications Origins of uncertainty • The variability of observed repeatable natural phenomena : « randomness ». – Coins, dice…: what about the outcome of the next throw? • The lack of information: incompleteness – because of information is often lacking, knowledge about issues of interest is generally not perfect. • Conflicting testimonies or reports:inconsistency – The more sources, the more likely the inconsistency Example • Variability: daily quantity of rain in Toulouse – May change every day – It can be estimated through statistical observed data. – Beliefs or prediction based on this data • Incomplete information : Birth date of Brazil President – It is not a variable: it is a constant! – You can get the correct info somewhere, but it is not available. – Most people may have a rough idea (an interval), a few know precisely, some have no idea: information is subjective. – Statistics on birth dates of other presidents do not help much. • Inconsistent information : several sources of information conflict concerning the birth date (a book, a friend, a website). The roles of probability Probability theory is generally used for representing two aspects: 1. Variability: capturing (beliefs induced by) variability through repeated observations. 2. Incompleteness (info gaps): directly modeling beliefs via betting behavior observation. These two situations are not mutually exclusive. Using a single probability distribution to represent incomplete information is not entirely satisfactory: The betting behavior setting of Bayesian subjective probability enforces a representation of partial ignorance based on single probability distributions.
    [Show full text]
  • Imprecise Probability for Multiparty Session Types in Process Algebra APREPRINT
    IMPRECISE PROBABILITY FOR MULTIPARTY SESSION TYPES IN PROCESS ALGEBRA A PREPRINT Bogdan Aman Gabriel Ciobanu Romanian Academy, Institute of Computer Science Romanian Academy, Institute of Computer Science Alexandru Ioan Cuza University, Ia¸si, Romania Alexandru Ioan Cuza University, Ia¸si, Romania [email protected] [email protected] February 18, 2020 ABSTRACT In this paper we introduce imprecise probability for session types. More exactly, we use a proba- bilistic process calculus in which both nondeterministic external choice and probabilistic internal choice are considered. We propose the probabilistic multiparty session types able to codify the structure of the communications by using some imprecise probabilities given in terms of lower and upper probabilities. We prove that this new probabilistic typing system is sound, as well as several other results dealing with both classical and probabilistic properties. The approach is illustrated by a simple example inspired by survey polls. Keywords Imprecise Probability · Nondeterministic and Probabilistic Choices · Multiparty Session Types. 1 Introduction Aiming to represent the available knowledge more accurately, imprecise probability generalizes probability theory to allow for partial probability specifications applicable when a unique probability distribution is difficult to be identified. The idea that probability judgments may be imprecise is not novel. It would be enough to mention the Dempster- Shafer theory for reasoning with uncertainty by using upper and lower probabilities [1]. The idea appeared even earlier in 1921, when J.M. Keynes used explicit intervals for approaching probabilities [2] (B. Russell called this book “undoubtedly the most important work on probability that has appeared for a very long time”).
    [Show full text]
  • Demystifying Dilation
    Forthcoming in Erkenntnis. Penultimate version. Demystifying Dilation Arthur Paul Pedersen Gregory Wheeler Abstract Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics main- tain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. How- ever, two points are often overlooked: (i) knowing that E is stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (ii) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence be- tween dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. 1 Good Grief! Unlike free advice, which can be a real bore to endure, accepting free information when it is available seems like a Good idea. In fact, it is: I. J. Good (1967) showed that under certain assumptions it pays you, in expectation, to acquire new infor- mation when it is free. This Good result reveals why it is rational, in the sense of maximizing expected utility, to use all freely available evidence when estimating a probability.
    [Show full text]
  • An Imprecise SHAP As a Tool for Explaining the Class Probability Distributions Under Limited Training Data
    An Imprecise SHAP as a Tool for Explaining the Class Probability Distributions under Limited Training Data Lev V. Utkin, Andrei V. Konstantinov, and Kirill A. Vishniakov Peter the Great St.Petersburg Polytechnic University St.Petersburg, Russia e-mail: [email protected], [email protected], [email protected] Abstract One of the most popular methods of the machine learning prediction explanation is the SHapley Additive exPlanations method (SHAP). An imprecise SHAP as a modification of the original SHAP is proposed for cases when the class probability distributions are imprecise and represented by sets of distributions. The first idea behind the imprecise SHAP is a new approach for computing the marginal contribution of a feature, which fulfils the important efficiency property of Shapley values. The second idea is an attempt to consider a general approach to calculating and reducing interval-valued Shapley values, which is similar to the idea of reachable probability intervals in the imprecise probability theory. A simple special implementation of the general approach in the form of linear optimization problems is proposed, which is based on using the Kolmogorov-Smirnov distance and imprecise contamination models. Numerical examples with synthetic and real data illustrate the imprecise SHAP. Keywords: interpretable model, XAI, Shapley values, SHAP, Kolmogorov-Smirnov distance, imprecise probability theory. arXiv:2106.09111v1 [cs.LG] 16 Jun 2021 1 Introduction An importance of the machine learning models in many applications and their success in solv- ing many applied problems lead to another problem which may be an obstacle for using the models in areas where their prediction accuracy as well as understanding is crucial, for ex- ample, in medicine, reliability analysis, security, etc.
    [Show full text]
  • Propagation of Imprecise Probabilities Through
    PROPAGATION OF IMPRECISE PROBABILITIES THROUGH BLACK-BOX MODELS A Thesis Presented to The Academic Faculty by Morgan Bruns In Partial Fulfillment of the Requirements for the Degree Master of Science in the School of Mechanical Engineering Georgia Institute of Technology May 2006 PROPAGATION OF IMPRECISE PROBABILITIES THROUGH BLACK-BOX MODELS Approved by: Dr. Christiaan J.J. Paredis, Advisor School of Mechanical Engineering Georgia Institute of Technology Dr. Bert Bras School of Mechanical Engineering Georgia Institute of Technology Dr. Leon F. McGinnis School of Industrial and Systems Engineering Georgia Institute of Technology Date Approved: April 7, 2006 ACKNOWLEDGEMENTS This thesis would not have been possible without the assistance of many of my friends and colleagues. I would like to thank my advisor, Dr. Chris Paredis, for his support, guidance, and insight. I would also like to thank the students of the Systems Realization Laboratory for the intellectually nurturing environment they have provided. In particular, I would like to thank Jason Aughenbaugh, Scott Duncan, Jay Ling, Rich Malak, and Steve Rekuc for their criticism and advice regarding my research. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii LIST OF ABBREVIATIONS viii LIST OF SYMBOLS ix SUMMARY xi CHAPTER 1 : INTRODUCTION 1 1.1 Design decision making 1 1.2 Imprecision in design 2 1.3 Challenges for decision making formalisms 4 1.4 Research questions, hypotheses, and thesis outline 5 CHAPTER 2 : DECISION MAKING UNDER IMPRECISE UNCERTAINTY
    [Show full text]
  • Demystifying Dilation
    CORE Metadata, citation and similar papers at core.ac.uk Provided by PhilSci Archive Forthcoming in Erkenntnis. Penultimate version. Demystifying Dilation Arthur Paul Pedersen Gregory Wheeler Abstract Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics main- tain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. How- ever, two points are often overlooked: (i) knowing that E is stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (ii) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence be- tween dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. 1 Good Grief! Unlike free advice, which can be a real bore to endure, accepting free information when it is available seems like a Good idea. In fact, it is: I. J. Good (1967) showed that under certain assumptions it pays you, in expectation, to acquire new infor- mation when it is free.
    [Show full text]
  • For Peer Review 19 20 2009)
    Cambridge Journal of Economics De Finetti on Uncertainty Journal: Cambridge Journal of Economics For PeerManuscript ID:Review Draft Manuscript Type: Article CJE Keywords: risk, uncertainty, de Finetti, Keynes, Knight <a href=http://www.aeaweb.org/journal/elclasjn_hold.html B21, D81 target=new><b>JEL Classification</b></a>: http://mc.manuscriptcentral.com/cje Page 1 of 29 Cambridge Journal of Economics 1 2 3 4 DE FINETTI ON UNCERTAINTY 5 6 7 8 9 10 11 Abstract 12 13 The well-known Knightian distinction between quantifiable risk and unquantifiable uncertainty is at 14 odds with the dominant subjectivist conception of probability associated with de Finetti, Ramsey 15 and Savage. Risk and uncertainty are rendered indistinguishable on the subjectivist approach insofar 16 as an individual’s subjective estimate of the probability of any event can be elicited from the odds at 17 which she would be prepared to bet for or against that event. The risk/uncertainty distinction has 18 however never quite goneFor away andPeer is currently undeReviewr renewed theoretical scrutiny. The purpose of 19 20 this paper is to show that de Finetti’s understanding of the distinction is more nuanced than is 21 usually admitted. Relying on usually overlooked excerpts of de Finetti’s works commenting on 22 Keynes, Knight, and interval valued probabilities, we argue that de Finetti suggested a relevant 23 theoretical case for uncertainty to hold even when individuals are endowed with subjective 24 probabilities. Indeed, de Finetti admitted that the distinction between risk and uncertainty is relevant 25 when different people sensibly disagree about the probability of the occurrence of an event.
    [Show full text]
  • Design Optimization with Imprecise Random Variables
    2009-01-0201 Design Optimization with Imprecise Random Variables Jeffrey W. Herrmann University of Maryland, College Park Copyright © 2009 SAE International ABSTRACT using precise probability distributions or representing uncertain quantities as fuzzy sets. Haldar and Design optimization is an important engineering design Mahadevan [7] give a general introduction to reliability- activity. Performing design optimization in the presence based design optimization, and many different solution of uncertainty has been an active area of research. The techniques have been developed [8, 9, 10]. Other approaches used require modeling the random variables approaches include evidence-based design optimization using precise probability distributions or representing [11], possibility-based design optimization [12], and uncertain quantities as fuzzy sets. This work, however, approaches that combine possibilities and probabilities considers problems in which the random variables are [13]. Zhou and Mourelatos [11] discussed an evidence described with imprecise probability distributions, which theory-based design optimization (EBDO) problem. are highly relevant when there is limited information They used a hybrid approach that first solves a RBDO to about the distribution of a random variable. In particular, get close to the optimal solution and then generates this paper formulates the imprecise probability design response surfaces for the active constraints and uses a optimization problem and presents an approach for derivative-free optimizer to find a solution. solving it. We present examples for illustrating the approach. The amount of information available and the outlook of the decision-maker (design engineer) determines the INTRODUCTION appropriateness of different models of uncertainty. No single model should be considered universally valid. In Design optimization is an important engineering design this paper, we consider situations in which there is activity in automotive, aerospace, and other insufficient information about the random variables to development processes.
    [Show full text]
  • Blurring out Cosmic Puzzles
    Blurring Out Cosmic Puzzles Yann Ben´etreau-Dupin∗† September 12, 2018 Abstract The Doomsday argument and anthropic reasoning are two puz- zling examples of probabilistic confirmation. In both cases, a lack of knowledge apparently yields surprising conclusions. Since they are for- mulated within a Bayesian framework, they constitute a challenge to Bayesianism. Several attempts, some successful, have been made to avoid these conclusions, but some versions of these arguments cannot be dissolved within the framework of orthodox Bayesianism. I show that adopting an imprecise framework of probabilistic reasoning allows for a more adequate representation of ignorance in Bayesian reasoning and explains away these puzzles. 1 Introduction The Doomsday paradox and the appeal to anthropic bounds to solve the cosmological constant problem are two examples of puzzles of probabilistic confirmation. These arguments both make ‘cosmic’ predictions: the former gives us a probable end date for humanity, and the second a probable value of the vacuum energy density of the universe. They both seem to allow one to draw unwarranted conclusions from a lack of knowledge, and yet one way of formulating them makes them a straightforward application of Bayesianism. They call for a framework of inductive logic that allows one to represent arXiv:1412.4382v1 [physics.data-an] 14 Dec 2014 ∗Department of Philosophy & Rotman Institute of Philosophy, Western University, Canada, [email protected] †Forthcoming in Philosophy of Science (PSA 2014). This paper stems from discussions at the UCSC-Rutgers Institute for the Philosophy of Cosmology in the summer of 2013. I am grateful to Chris Smeenk and Wayne Myrvold for discussions and comments on earlier drafts.
    [Show full text]
  • On De Finetti's Instrumentalist Philosophy of Probability
    Forthcoming in the European Journal for Philosophy of Science On de Finetti’s instrumentalist philosophy of probability Joseph Berkovitz1 Abstract. De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. 1. Introduction De Finetti is one of the founding fathers of the modern subjective school of probability. In this school of thought, probabilities are commonly conceived as coherent degrees of belief, and expectations are defined in terms of such probabilities. By contrast, de Finetti took expectation as the fundamental concept of his theory of probability and subjective probability as derivable from expectation. In his theory, probabilities are coherent degrees of expectation.2 De Finetti represents the radical wing of the subjective school of probability, denying the existence of objective probabilities. He held that probabilities and probabilistic reasoning are subjective and that none of the objective interpretations of probability make sense. Accordingly, he rejected the idea that subjective probabilities are guesses, predictions, or hypotheses about objective probabilities.
    [Show full text]
  • Reasoning with Imprecise Belief Structures
    Reasoning with imprecise belief structures Thierry Denœux U.M.R. CNRS 6599 Heudiasyc Universit´ede Technologie de Compi`egne BP 20529 - F-60205 Compi`egne cedex - France email: Thierry.Denoeux@@hds.utc.fr Abstract This paper extends the theory of belief functions by introducing new concepts and techniques, allowing to model the situation in which the be- liefs held by a rational agent may only be expressed (or are only known) with some imprecision. Central to our approach is the concept of interval- valued belief structure, defined as a set of belief structures verifying certain constraints. Starting from this definition, many other concepts of Evidence Theory (including belief and plausibility functions, pignistic probabilities, combination rules and uncertainty measures) are generalized to cope with imprecision in the belief numbers attached to each hypothesis. An appli- cation of this new framework to the classification of patterns with partially known feature values is demonstrated. Keywords: evidence theory, Dempster-Shafer theory, belief functions, Transferable Belief model, uncertainty modeling, imprecision, pattern clas- sification. 1 Introduction The representation of uncertainty is a major issue in many areas of Science and Engineering. Although Probability Theory has been regarded as a reference framework for about two centuries, it has become increasingly apparent that the concept of probability is not general enough to account for all kinds of uncertainty. More specifically, the requirement that precise numbers be assigned to every individual hypotheses is often regarded as too restrictive, particularly when the available information is poor. The use of completely monotone capacities, also called credibility or belief functions, as a general tool for representing someone's degrees of belief was proposed by Shafer [20] in the seventies (belief functions had already been introduced by Dempster ten years earlier as lower probabilities generated by a multivalued mapping [1]).
    [Show full text]
  • Mean and Variance Bounds and Propagation for Ill-Specified Random Variables Andrew T
    494 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 34, NO. 4, JULY 2004 Mean and Variance Bounds and Propagation for Ill-Specified Random Variables Andrew T. Langewisch and F. Fred Choobineh, Member, IEEE Abstract—Foundations, models, and algorithms are provided Furthermore, we show that nonexhaustive sensitivity analysis for identifying optimal mean and variance bounds of an ill-spec- does not always identify the optimal bounds for the mean and ified random variable. A random variable is ill-specified when at variance. least one of its possible realizations and/or its respective proba- bility mass is not restricted to a point but rather belongs to a set Ill-specificity is a consequence of imprecision or ambi- or an interval. We show that a nonexhaustive sensitivity-analysis guity concerning random variables, and in situations where approach does not always identify the optimal bounds. Also, a pro- arithmetic functions of several ill-specified variables are of cedure for determining the mean and variance bounds of an arith- interest, researchers have examined various techniques which metic function of ill-specified random variables is presented. Esti- model the resultant distributional uncertainty and imprecision. mates of pairwise correlation among the random variables can be incorporated into the function. The procedure is illustrated in the These techniques include robust Bayesian methods (e.g., [1]), context of a case study in which exposure to contaminants through Monte Carlo simulation with best judgment to choose input the inhalation pathway is modeled. distributions (e.g., [2], [3]), Monte Carlo simulation with the Index Terms—Fuzzy numbers, ill-specified random variables, input distributions selected by the maximum entropy criterion mean and variance bounds, risk analysis, uncertainty.
    [Show full text]