updating beliefs with incomplete observations currently there is renewed interest in the problem raised by shafer in 1985 of updating probabilities when observations are incomplete or set valued this is a fundamental problem in general and of particular interest for bayesian networks recently grunwald and halpern have shown that commonly used updating strategies fail in this case except under very special assumptions in this paper we propose a new method for updating probabilities with incomplete observations our approach is deliberately conservative we make no assumptions about the so called incompleteness mechanism that associates complete with incomplete observations we model our ignorance about this mechanism by a vacuous lower prevision a tool from the theory of imprecise probabilities and we use only coherence arguments to turn prior into posterior updated probabilities in general this new approach to updating produces lower and upper posterior probabilities and previsions expectations as well as partially determinate decisions this is a logical consequence of the existing ignorance about the incompleteness mechanism as an example we use the new updating method to properly address the apparent paradox in the monty hall puzzle more importantly we apply it to the problem of classification of new evidence in probabilistic expert systems where it leads to a new so called conservative updating rule in the special case of bayesian networks constructed using expert knowledge we provide an exact algorithm to compare classes based on our updating rule which has linear time complexity for a class of networks wider than polytrees this result is then extended to the more general framework of credal networks where computations are often much harder than with bayesian nets using an example we show that our rule appears to provide a solid basis for reliable updating with incomplete observations when no strong assumptions about the incompleteness mechanism are justified

upper and lower conditional probabilities induced by a multivalued mapping abstract given a finitely additive full conditional probability space x f f 0 and a conditional measurable space y g g 0 a multivalued mapping from x to y induces a class of full conditional probabilities on y g g 0 a closed form expression for the lower and upper envelopes and of such class is provided the envelopes can be expressed through a generalized bayesian conditioning rule relying on two linearly ordered classes of possibly unbounded inner and outer measures for every b g 0 b is a normalized totally monotone capacity which is continuous from above if x f f 0 is a countably additive full conditional probability space and f is a algebra moreover the full conditional prevision functional m induced by on the set of f continuous conditional gambles is shown to give rise through to the lower and upper full conditional prevision functionals m and m on the set of g continuous conditional gambles for every b g 0 m b is a totally monotone functional having a choquet integral expression involving finally by considering another conditional measurable space z h h 0 and a multivalued mapping from y to z it is shown that the conditional measures and functionals m m induced by preserve the same properties of and m m

higher order models for fuzzy random variables a fuzzy random variable is viewed as the imprecise observation of the outcomes in a random experiment since randomness and vagueness coexist in the same framework it seems reasonable to integrate fuzzy random variables into imprecise probabilities theory nevertheless fuzzy random variables are commonly presented in the literature as classical measurable functions associated to a classical probability measure we present here a higher order possibility model that represents the imprecise information provided by a fuzzy random variable we compare it with previous classical models in the literature first some aspects about the acceptability function associated to a fuzzy random variable are investigated secondly we present three different higher order possibility models all of them arising in a natural way we investigate their similarities and differences and observe that the first one the fuzzy probability envelope is the most informative finally we compare the fuzzy probability envelope with the classical probability measure induced by the fuzzy random variable we conclude that the classical probability measure does not always contain all relevant information provided by a fuzzy random variable

eliciting dual interval probabilities from interval comparison matrices in recent years dealing with uncertainty using interval probabilities such as combination marginalization condition bayesian inferences has received considerable attention however how to elicit interval probabilities from subjective judgment is still a fundamental problem in applications of interval probabilities in this paper interval valued pairwise comparisons of events are used to characterize the judgment of a person about which one is more likely to occur between each pair of events from the interval comparison matrix we elicit the dual interval probabilities by linear programming problems the ratios of the estimated interval probabilities are used to approximate the interval comparison values from the outside and the inside of them in other words the inclusion relations between the interval comparison values and the ratios of the obtained interval probabilities hold a numerical example concerning a new product development is presented to illustrate how to estimate dual interval probabilities and use interval probabilities to make a decision

the consensus operator for combining beliefs the consensus operator provides a method for combining possibly conflicting beliefs within the dempster shafer belief theory and represents an alternative to the traditional dempster s rule this paper describes how the consensus operator can be applied to dogmatic conflicting opinions i e when the degree of conflict is very high it overcomes shortcomings of dempster s rule and other operators that have been proposed for combining possibly conflicting beliefs

on the granularity of summative kernels in this paper we propose granularity as a new index to characterize the non specificity of a summative kernel this index is intended to reflect the behavior of a kernel in the usual signal processing applications we show in different experiments that two kernels having the same granularity have very similar behavior this granularity based adaptation is compared to other adaptation methods these experiments highlight the ability of the granularity index to measure the spreading and collecting properties of a summative kernel

inferential models a framework for prior free posterior probabilistic inference posterior probabilistic statistical inference without priors is an important but so far elusive goal fisher s fiducial inference dempster shafer theory of belief functions and bayesian inference with default priors are attempts to achieve this goal but to date none has given a completely satisfactory picture this article presents a new framework for probabilistic inference based on inferential models ims which not only provides data dependent probabilistic measures of uncertainty about the unknown parameter but also does so with an automatic long run frequency calibration property the key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter and the prediction of this auxiliary variable with a random set before conditioning on data here we present a three step im construction and prove a frequency calibration property of the im s belief function under mild conditions a corresponding optimality theory is develo

distributional properties of correspondences on loeb spaces abstract we present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a loeb space which include closedness convexity compactness purification and semicontinuity we also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit lebesgue interval

on the connection between probability boxes and possibility measures we explore the relationship between possibility measures supremum preserving normed measures and p boxes pairs of cumulative distribution functions on totally preordered spaces extending earlier work in this direction by de cooman and aeyels among others we start by demonstrating that only those p boxes who have 0 1 valued lower or upper cumulative distribution function can be possibility measures and we derive expressions for their natural extension in this case next we establish necessary and sufficient conditions for a p box to be a possibility measure finally we show that almost every possibility measure can be modelled by a p box simply by ordering elements by increasing possibility whence any techniques for p boxes can be readily applied to possibility measures we demonstrate this by deriving joint possibility measures from marginals under varying assumptions of independence using a technique known for p boxes doing so we arrive at a new rule of combination for possibility measures for the independent case

supremum preserving upper probabilities abstract we study the relation between possibility measures and the theory of imprecise probabilities and argue that possibility measures have an important part in this theory it is shown that a possibility measure is a coherent upper probability if and only if it is normal a detailed comparison is given between the possibilistic and natural extension of an upper probability both in the general case and for upper probabilities defined on a class of nested sets we prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability defined on a class of nested sets we show that possibilistic extension can be interpreted in terms of natural extension we also prove that when either the upper or the lower cumulative distribution function of a random quantity is specified possibility measures very naturally emerge as the corresponding natural extensions next we go from upper probabilities to upper previsions we show that if a coherent upper prevision defined on the convex cone of all non negative gambles is supremum preserving then it must take the form of a shilkret integral associated with a possibility measure but at the same time we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a non empty subset of the universe of discourse

n monotone exact functionals we study n monotone functionals which constitute a generalisation of n monotone set functions we investigate their relation to the concepts of exactness and natural extension which generalise coherence and natural extension in the behavioural theory of imprecise probabilities we improve upon a number of results in the literature and prove among other things a representation result for exact n monotone functionals in terms of choquet integrals

augmented lagrangian finite elements for plate contact problems the present work investigates the unilateral frictionless contact between a plate and a rigid obstacle two sets of problems are studied a plate constrained through unilateral edge supports and a plate seating in its undeformed configuration at a given distance from a rigid support the attention is concentrated on two augmented lagrangian formulations the algorithmic implementation within a finite element scheme is presented and discussed the importance of using appropriate plate elements for the discretization of the structure is stressed new gap elements compatible with a robust plate element are derived computational aspects are emphasized a simple and effective numerical integration for the determination of the gap stiffnesses in partial contact with the support is proposed numerical results are carried out and compared with analytical solutions the convergence to the solution of the perfectly constrained problem is numerically investigated the inadequacy of the penalty method and the satisfactory performance obtained from only one augmented lagrangian procedure are emphasized

a unifying approach to integration for bounded positive charges this paper deals with n monotone functionals which constitute a generalisation of n monotone set functions using the notion of exactness of a functional we introduce a new notion of lower and upper integral which subsumes as particular cases most of the approaches to integration in the literature as a consequence we can characterise which types of integrals can be used to calculate the natural extension the lower envelope of all linear extensions of a positive bounded charge

the prevision of a prevision abstract we prove that the result ex e e x y is true for bounded x when the usual concept of conditional expectation or prevision is replaced by an alternative definition reflecting an individual s actual beliefs concerning x after observing y we discuss the importance of this result to subjectivist theory

fuzzy integral representation abstract in this paper we give representation results for nonlinear functionals in arbitrary spaces as fuzzy integrals the main assumption is that of comonotonic maxitivity i e i f g if ig for comonotonic f g under the more restrictive condition of stochastic dominance we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure we discuss the particular cases of possibility measures and upper fuzzy expectations some partial randon nikodym results are proved for both the choquet and the fuzzy integrals finally an example is shown where the fuzzy integral representation is useful

conditional belief functions as lower envelopes of conditional probabilities in a finite setting the aim is to provide a characterization of full conditional measures on a finite boolean algebra obtained as lower envelope of the extensions of a full conditional probability defined on another finite boolean algebra such conditional measures are conditional belief functions defined by means of a generalized bayesian conditioning rule relying on a linearly ordered class of belief functions this notion of bayesian conditioning for belief functions is compared with other well known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities

probability and time probabilistic reasoning is often attributed a temporal meaning in which conditioning is regarded as a normative rule to compute future beliefs out of current beliefs and observations however the well established updating interpretation of conditioning is not concerned with beliefs that evolve in time and in particular with future beliefs on the other hand a temporal justification of conditioning was proposed already by de moivre and bayes by requiring that current and future beliefs be consistent we reconsider the latter approach while dealing with a generalised version of the problem using a behavioural theory of imprecise probability in the form of coherent lower previsions as well as of coherent sets of desirable gambles and letting the possibility space be finite or infinite we obtain that using conditioning is normative in the imprecise case only if one establishes future behavioural commitments at the same time of current beliefs in this case it is also normative that present beliefs be conglomerable which is a result that touches on a long term controversy at the foundations of probability in the remaining case where one commits to some future behaviour after establishing present beliefs we characterise the several possibilities to define consistent future assessments this shows in particular that temporal consistency does not preclude changes of mind and yet our analysis does not support that rationality requires consistency in general even though pursuing consistency makes sense and is useful at least as a way to guide and evaluate the assessment process these considerations narrow down in the special case of precise probability because this formalism cannot distinguish the two different situations illustrated above it turns out that the only consistent rule is conditioning and moreover that it is not rational to be willing to stick to precise probability while using a rule different from conditioning to compute future beliefs rationality requires in addition the disintegrability of the present time probability

a behavioral model for linguistic uncertainty abstract the paper discusses the problem of modelling linguistic uncertainty which is the uncertainty produced by statements in natural language for example the vague statement mary is young produces uncertainty about mary s age we concentrate on simple affirmative statements of the type subject is predicate where the predicate satisfies a special condition called monotonicity for this case we model linguistic uncertainty in terms of upper probabilities which are given a behavioural interpretation as betting rates possibility measures and probability measures are special types of upper probability measure we evaluate zadeh s suggestion that possibility measures should be used to model linguistic uncertainty and the bayesian claim that probability measures should be used our main conclusion is that when the predicate is monotonic possibility measures are appropriate models for linguistic uncertainty we also discuss several assessment strategies for constructing a numerical model

approximations of upper and lower probabilities by measurable selections a random set can be regarded as the result of the imprecise observation of a random variable following this interpretation we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable we link this problem to the existence of selectors of a multi valued mapping and with the inner approximations of the upper probability and prove that under fairly general conditions although not in all cases the upper and lower probabilities are an adequate tool for modelling the available information in doing this we generalise a number of results from the literature finally we study the particular case of consonant random sets and we also derive a relationship between aumann and choquet integrals

bayes theorem for choquet capacities we give an upper bound for the posterior probability of a measurable set a when the prior lies in a class of probability measures p the bound is a rational function of two choquet integrals if p is weakly compact and is closed with respect to majorization then the bound is sharp if and only if the upper prior probability is 2 alternating the result is used to compute bounds for several sets of priors used in robust bayesian inference the result may be regarded as a characterization of 2 alternating choquet capacities

weak sequential convergence in l1 μ x and an exact version of fatouʼs lemma abstract the class of nonatomic finite measure spaces with the saturation property as developed in maharam 1942 and hoover keisler 1984 is characterized by the fatou and lebesgue property of a well dominated sequence of multifunctions taking values in a banach space with multifunctions reduced to functions this fatou characterization also extends to a variant of the closure property found in optimal control theory the results are developed through a considered overview of the relevant literature on the exact and approximate fatou lemma phrased in terms of bochner integration

coherent lower previsions and choquet integrals walley statistical reasoning with imprecise probabilities chapman hall london 1991 has given a helpful general framework of uncertain reasoning which is grounded on a behavioural foundation and extends de finetti s work for some important applications like the expected utility theory it might be a tool alternatively to the techniques of choquet integration therefore this paper tries to clarify the relation between walley s central concept of coherent lower prevision and the choquet integration theory in one direction we deal with the problem to represent coherent lower previsions by choquet integrals and obtain general results using greco s representation theorem another aspect of investigation is to compare walley s concept to define integrals of coherent fuzzy measures the so called natural extensions with choquet integrals we succeed in exploring completely the relationship

a random set characterization of possibility measures several authors have pointed out the relationship between consonant random sets and possibility measures however this relationship has only been proven for the finite case where the inverse mobius of the upper probability induced by the random set simplifies the computations to a great extent in this paper we study the connection between both concepts for arbitrary referential spaces we complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases

adapted probability distributions on introduit une famille de notions d equivalence pour des processus stochastiques sur des espaces sous tendus par une filtration

independent random partial matching with general types abstract we provide a mathematical foundation for an independent random partial matching with a continuum of agents where the type space is not necessarily finite besides establishing the existence result we also identify the deterministic distributions of matched types in the matching our results are based on the fubini property for loeb product spaces and products of loeb transition probabilities as well as keisler s homogeneity theorem for loeb counting probability spaces

a probabilistic representation of exact games on σ algebras the purpose of this paper is to establish the intrinsic relations between the cores of exact games on s algebras and the extensions of exact games to function spaces given a probability space exact functionals are defined on l as an extension of exact games to derive a probabilistic representation for exact functionals we endow them with two probabilistic conditions law invariance and the fatou property the representation theorem for exact functionals lays a probabilistic foundation for nonatomic scalar measure games based on the notion of p convexity we also investigate the equivalent conditions for the representation of anonymous convex games

a general fatou lemma a general fatou lemma is established for a sequence of gelfand integrable functions from a vector loeb space to the dual of a separable banach space or with a weaker assumption on the sequence a banach lattice a corollary sharpens previous results in the finite dimensional setting even for the case of scalar measures counterexamples are presented to show that the results obtained here are sharp in various aspects applications include systematic generalizations of the distribution of correspondences from the case of scalar loeb spaces to the case of vector loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces

lower previsions induced by filter maps coherent lower previsions are amongst the most prominent uncertainty models within imprecise probability theory we show how they can model information about the joint behaviour of two variables when these are related by means of a filter map a model for the imprecise observation of a random variable by means of a class of filters our construction preserves a number of interesting properties such as n monotonicity and it generalises a number of existing results for multi valued mappings

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

a geometric and game theoretic study of the conjunction of possibility measures we study the conjunction of possibility measures within imprecise probability theory we identify conditions under which the conjunction remains a possibility measure we provide a graphical way to check these conditions using zero sum games we provide a graphical way to adjust possibility measures so their conjunction is a possibility measure we identify conditions under which the conjunction is coherent in this paper we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities that is as upper bounds for some set of probability measures we identify conditions under which the minimum of two possibility measures remains a possibility measure we provide graphical way to check these conditions by means of a zero sum game formulation of the problem this also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure finally we identify conditions under which the minimum of two possibility measures is a coherent upper probability or in other words conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures

when upper probabilities are possibility measures abstract a characteristic property is given for a pair of upper and lower probabilities induced by lower probability bounds on a finite set of events to coincide with possibility and necessity measures approximations of upper probabilities by possibility measures are discussed the problem of combining possibility distributions viewed as upper probabilities is investigated and the basic fuzzy set intersections are justified in this framework

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

possibility and necessity integrals abstract in this paper we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures these confidence measures have a field of sets as their domain and a complete lattice as their codomain in introducing these integrals the analogy with the classical introduction of lebesgue integrals is explored and exploited it is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties in this way our dual classes of fuzzy integrals constitute a significant generalization of sugeno s fuzzy integrals a large number of important general properties of these integrals is studied furthermore and most importantly the combination of seminormed fuzzy integrals and possibility measures on the one hand and semiconormed fuzzy integrals and necessity measures on the other hand is extensively studied it is shown that these combinations are very natural and have properties which are analogous to the combination of lebesgue integrals and classical measures using these results the very basis is laid for a unifying measure and integral theoretic account of possibility and necessity theory in very much the same way as the theory of lebesgue integration provides a proper framework for a unifying and formal account of probability theory

prior envelopes based on belief functions we show that the mathematical structure of belief functions makes them suitable for generating classes of prior distributions to be used in robust bayesian inference in particular the upper and lower bounds of the posterior probability content of a measurable subset of the parameter space may be calculated directly in terms of upper and lower expectations we also extend an integral representation given by dempster to infinite sets

representing parametric probabilistic models tainted with imprecision numerical possibility theory belief functions have been suggested as useful tools to represent imprecise vague or incomplete information they are particularly appropriate in uncertainty analysis where information is typically tainted with imprecision or incompleteness based on their experience or their knowledge about a random phenomenon experts can sometimes provide a class of distributions without being able to precisely specify the parameters of a probability model frequentists use two dimensional monte carlo simulation to account for imprecision associated with the parameters of probability models they hence hope to discover how variability and imprecision interact this paper presents the limitations and disadvantages of this approach and propose a fuzzy random variable approach to treat this kind of knowledge

symmetric upper probabilities as a first step toward developing statistical models based on upper and lower probabilities we study upper probabilities and upper expectations on the unit interval that are symmetric by which we mean invariant with respect to equimeasurability these upper probabilities are generalizations of uniform probability measures we give some characterizations of these upper probabilities specifically we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization we also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star shaped with respect to the origin and to the point 1 1 we derive inner and outer approximations to symmetric classes of probabilities based on the upper probability the class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized we provide a geometric characterization of a hierarchy of upper probabilities including fine s generalized upper probabilities and 2 alternating choquet capacities in particular we establish a 1 1 correspondence between symmetric 2 alternating capacities and nonincreasing density functions we prove that undominated generalized upper probabilities do not exist in the symmetric case examples from robust statistics are considered an example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations a corollary is that symmetry of the choquet integral does not imply symmetry of the upper expectation functional

integration of correspondences on loeb spaces we study the bochner and gel fand integration of banach space valued correspondences on a general loeb space though it is well known that the lyapunov type result on the compactness and convexity of the integral of a correspondence and the fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space we show that exact versions of these two results hold in the case we study we also note that our results on a hyperfinite loeb space capture the nature of the corresponding asymptotic results for the large finite case but the unit lebesgue interval fails to provide such a framework

a semantics for possibility theory based on likelihoods abstract in this paper a semantic basis for possibility theory based on likelihood functions is presented in some cases possibilities have been considered as approximations of shafer plausibility measures this approximation exchanges exactness of plausibility values for the simplicity of use of possibility values in this paper a different direction is followed possibility measures are considered as the supremum of a family of likelihood functions this is an exact interpretation not an approximation the minimum rule to combine possibility distributions is justified in this framework under general conditions conditions under which other rules can be applied are also studied

a shear deformable plate element with an exact thin limit abstract we present a quadrilateral finite element developed within the framework of a shear deformable plate theory the interpolation for the rotation takes advantage of internal rotational degrees of freedom through the use of bubble functions while the interpolation for the transverse displacement is linked to the nodal rotations a careful study of the element behavior is performed using an extensive set of mixed patch tests results from several numerical examples are also presented the element has proper rank and excellent interpolating capacity moreover without using any ad hoc assumption e g energy balancing schemes the element presents no locking effects at all in fact the shear energy may be set identically equal to zero without introducing any ill conditioning in the problem thus recovering a proper thin plate limit

representation theorems for partially exchangeable random variables we provide representation theorems for both finite and countable sequences of finite valued random variables that are considered to be partially exchangeable in their most general form our results are presented in terms of sets of desirable gambles a very general framework for modelling uncertainty its key advantages are that it allows for imprecision is more expressive than almost every other imprecise probabilistic framework and makes conditioning on events with lower probability zero non problematic we translate our results to more conventional although less general frameworks as well lower previsions linear previsions and probability measures the usual precise probabilistic representation theorems for partially exchangeable random variables are obtained as special cases

a unifying note on fatou s lemma in several dimensions a general version of fatou s lemma in several dimensions is presented it subsumes the fatou lemmas given by schmeidler schmeidler d 1970 fatou s lemma in several dimensions proc amer math soc 24 300 306 hildenbrand hildenbrand w 1974 core and equilibria of a large economy princeton university press princeton cesari suryanarayana cesari l m b suryanarayana 1978 an existence theorem for pareto problems nonlinear anal 2 225 233 and artstein artstein z 1979 a note on fatou s lemma in several dimensions j math econom 6 277 282 also it is equivalent to an abstract variational existence result that extends and generalizes results by aumann perles aumann r j m perles 1965 a variational problem arising in economics j math anal appl 11 488 503 berliocchi lasry berliocchi h j m lasry 1973 integrands normales et mesures parametrees en calcul des variations bull soc math france101 129 184 artstein artstein z 1974 on a variational problem j math anal appl 45 404 415 and balder balder e j 1979 on a useful compactification for optimal control problems j math anal appl 72 391 398 in several respects

decision making with imprecise probabilities dempster shafer theory and application the information and data used to support decision making under uncertainty in water resources situations can often be characterized as being very limited or weak the representation of knowledge in conventional bayesian decision analysis is in the form of precisely specified distributions and is the same no matter how weak the information source for this knowledge a bayesian analysis therefore may inadvertently impart too much precision to the input information and to the results the concept of imprecise probability addresses this problem of excessive precision and a number of methods incorporating this concept have emerged one such method developed by dempster and shafer accommodates greater imprecision by allowing the specification of probabilities on intervals theoretical aspects of the dempster shafer methodology and its application to inference and decision analysis are described a water resources example of an application of the dempster shafer approach is presented and the results contrasted with those obtained from the closest equivalent bayesian scheme

set valued functions lebesgue extensions and saturated probability spaces abstract recent advances in the theory of distributions of set valued functions have been shaped by counterexamples which hinge on the non existence of measurable selections with requisite properties these examples all based on the lebesgue interval and initially circumvented by sun in the context of loeb spaces have now led keisler and sun ks to establish a comprehensive theory of the distributions of set valued functions on saturated probability spaces introduced by hoover and keisler in contrast we show that a countably generated extension of the lebesgue interval suffices for an explicit resolution of these examples and furthermore that it does not contradict the ks necessity results we draw the fuller implications of our theorems for integration of set valued functions for lyapunov s result on the range of vector measures and for the theory of large non anonymous games

constrained optimization problems under uncertainty with coherent lower previsions we investigate a constrained optimization problem with uncertainty about constraint parameters our aim is to reformulate it as a constrained optimization problem without uncertainty this is done by recasting the original problem as a decision problem under uncertainty we give results for a number of different types of uncertainty models linear and vacuous previsions and possibility distributions and for two common but different optimality criteria for such decision problems maximinity and maximality we compare our approach with other approaches that have appeared in the literature

fatou s lemma for sugeno integral abstract fatou s lemma plays an important role in classical probability and measure theory non additive measure is a generalization of additive probability measure sugeno s integral is a useful tool in several theoretical and applied statistics which have been built on non additive measure in this paper a fatou type lemma for sugeno integral is shown the studied inequality is based on the classical fatou lemma for lebesgue integral to illustrate the proposed inequalities some examples are given

quasi continuous histograms histograms are very useful for summarizing statistical information associated with a set of observed data they are one of the most frequently used density estimators due to their ease of implementation and interpretation however histograms suffer from a high sensitivity to the choice of both reference interval and bin width this paper addresses this difficulty by means of a fuzzy partition we propose a new density estimator based on transferring the counts associated with each cell of the fuzzy partition to any subset of the reference interval we introduce three different methods of achieving this transfer the properties of each method are illustrated with a classic real observation set the density estimator obtained relates to the parzen rosenblatt kernel density estimation technique in this paper we only consider the monovariate case with precise and imprecise observations

independent products in infinite spaces abstract probabilistic independence intended as the mutual irrelevance of given variables can be solidly founded on a notion of self consistency of an uncertainty model in particular when probabilities go imprecise there is nothing in this approach that prevents it from being adopted in very general setups and yet it has mostly been detailed for variables taking finitely many values in this mathematical study we complement previous research by exploring the extent to which such an approach can be generalised we focus in particular on the independent products of two variables we characterise the main notions including some of factorisation and productivity in the general case where both spaces can be infinite and show that however there are situations even in the case of precise probability where no independent product exists this is not the case as soon as at least one space is finite we study in depth this case at the frontiers of good behaviour detailing the relations among the most important notions we show for instance that being an independent product is equivalent to a certain productivity condition then we step back to the general case we give conditions for the existence of independent products and study ways to get around its inherent limitations

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

robust bayes and empirical bayes analysis with _ epsilon contaminated priors on considere deux directions dans l analyse bayesienne robuste la premiere consiste a determiner l etendue des probabilites a posteriori d un ensemble la seconde consiste a selectionner une bonne distribution a priori

the significance of jacob bernoulli s ars conjectandi for the philosophy of probability today abstract more than 300 years ago in a fertile period from 1684 to 1689 jacob bernoulli worked out a strategy for applying the mathematics of games of chance to the domain of probability that strategy came into public view in august 1705 eight years after jacob s death at the age of 50 when his masterpiece ars conjectandi was published unfortunately the message of ars conjectandi was not fully absorbed at the time of its publication and it has been obscured by various intellectual fashions during the past 300 years the theme of this article is that jacob s contributions to the philosophy of probability are still fresh in many ways after 300 years we still have something to learn from his way of seeing the problems and from his approach to solving them and the fog that obscured his ideas is clearing our intellectual situation today may permit us a deeper appreciation of jacob s message than any of our predecessors were able to achieve i will pursue this theme chronologically first i will review jacob s problem situation and how he dealt with it second i will look at the changing obstacles to the reception of his message over the last three centuries finally i will make a case for paying more attention today

possibilistic and probabilistic likelihood functions and their extensions common features and specific characteristics we deal with conditional probability in the sense of de finetti and with t conditional possibility with t a triangular norm we prove that dubois and prade conditional possibility is a particular min conditional possibility and then we compare the two notions of conditioning by an inferential point of view moreover we study t conditional possibilities as functions of the conditioning event putting in evidence analogies and differences with conditional probabilities this allows to characterize likelihood functions and their aggregations consistent either with a t conditional possibility or a conditional probability this analysis highlights many syntactical coincidences nevertheless the main difference is a weak form of monotonicity which arises only in the possibilistic case

inferential processes leading to possibility and necessity this paper deals with the upper and lower bounds of a class of uncertainty measures endowed with particular characteristics decomposability monotonicity partial additivity and so on we consider an initial partial assessment consistent with either probability or possibility or necessity then we study the upper and lower envelopes of all possible extensions by resorting to a notion of weak logical independence we get as lower or upper envelope a possibility or a necessity respectively starting either from a probability or from a possibility or from a necessity

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

imprecise probability trees bridging two theories of imprecise probability we give an overview of two approaches to probability theory where lower and upper probabilities rather than probabilities are used walley s behavioural theory of imprecise probabilities and shafer and vovk s game theoretic account of probability we show that the two theories are more closely related than would be suspected at first sight and we establish a correspondence between them that i has an interesting interpretation and ii allows us to freely import results from one theory into the other our approach leads to an account of probability trees and random processes in the framework of walley s theory we indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees and we prove an interesting and quite general version of the weak law of large numbers

the necessity of nowhere equivalence abstract we prove some regularity properties convexity closedness compactness and preservation of upper hemicontinuity for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition we show the necessity of such a condition for any of these properties to hold as an application we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure strategy nash equilibria exist in general large games with any fixed uncountable compact action space

bayes s two arguments for the rule of conditioning the introductory section of thomas bayes s famous essay on probability contains two arguments for what we now call the rule of conditioning the first argument which leads to bayes s third proposition can be made rigorous if we use rooted trees to represent the step by step determination of events the second argument which leads to bayes s fifth proposition does not stand up to scrutiny

discussion further thoughts on possibilistic previsions a rejoinder this is a rejoinder to the comments moral scozzafava utkin zadeh fuzzy sets and systems 2005 by a number of authors on my paper about possibilistic previsions

indeterminate probabilities on finite sets this paper presents a quasi bayesian model of subjective uncertainty in which beliefs which are represented by lower and upper probabilities qualified by numerical confidence weights the representation is derived from a system of axioms of binary preferences which differs from standard axiom systems insofar as completeness is not assumed and transitivity is weakened confidence weighted probabilities may be elicited through the acceptance of bets with limited stakes a generalization of the operational method of de finetti the model is applicable to the reconciliation of inconsistent probability judgments and to the sensitivity analysis of bayesian decision models

on maxitive integration abstract a functional is said to be maxitive if it commutes with the pointwise supremum operation such functionals find application in particular in decision theory and related fields in the present paper maxitive functionals are characterized as integrals with respect to maxitive measures also known as possibility measures or idempotent measures these maxitive integrals are then compared with the usual additive and nonadditive integrals on the basis of some important properties such as convexity subadditivity and the law of iterated expectations

a behavioural model for vague probability assessments i present an hierarchical uncertainty model that is able to represent vague probability assessments and to make inferences based on them this model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty and is based on walley s theory of imprecise probabilities it is formally closely related to zadeh s fuzzy probabilities but it has a different interpretation and a different calculus through rationality coherence arguments the hierarchical model is shown to lead to an imprecise first order uncertainty model that can be used in decision making and as a prior in statistical reasoning

measures of uncertainty in expert systems abstract this paper compares four measures that have been advocated as models for uncertainty in expert systems the measures are additive probabilities used in the bayesian theory coherent lower or upper previsions belief functions used in the dempster shafer theory and possibility measures fuzzy logic special emphasis is given to the theory of coherent lower previsions in which upper and lower probabilities expectations and conditional probabilities are constructed from initial assessments through a technique of natural extension mathematically all the measures can be regarded as types of coherent lower or upper previsions and this perspective gives some insight into the properties of belief functions and possibility measures the measures are evaluated according to six criteria clarity of interpretation ability to model partial information and imprecise assessments especially judgements expressed in natural language rules for combining and updating uncertainty and their justification consistency of models and inferences feasibility of assessment and feasibility of computations each of the four measures seems to be useful in special kinds of problems but only lower and upper previsions appear to be sufficiently general to model the most common types of uncertainty

computing bounds on expectations abstract one method for evaluating the sensitivity of a bayesian analysis is to embed the prior into a class of priors then bounds on prior and posterior quantities of interest must be computed this approach to inference often called robust bayesian inference has received much attention lately implementing robust bayesian methods entails difficult computations especially if the parameter space is high dimensional in this article we develop a monte carlo approach to computing these bounds and also explore some interesting theoretical properties of certain classes of priors the methods can be useful in other situations in which bounds on expectations are required