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

symmetric coherent choquet capacities choquet capacities are a generalization of probability measures that arise in robustness decision theory and game theory many capacities that arise in robustness are symmetric or can be transformed into sym metric capacities we characterize the extreme points of the set of upper distribution functions corresponding to coherent symmetric choquet ca pacities on 0 1 we also show that the set of 2 altemating capacities is a simplex and we give a choquet representation of this set

inferences from multinomial data learning about a bag of marbles a new method is proposed for making inferences from multinomial data in cases where there is no prior information a paradigm is the problem of predicting the colour of the next marble to be drawn from a bag whose contents are initially completely unknown in such problems we may be unable to formulate a sample space because we do not know what outcomes are possible this suggests an invariance principle inferences based on observations should not depend on the sample space in which the observations and future events of interest are represented objective bayesian methods do not satisfy this principle this paper describes a statistical model called the imprecise dirichlet model for drawing coherent inferences from multinomial data inferences are expressed in terms of posterior upper and lower probabilities the probabilities are initially vacuous reflecting prior ignorance but they become more precise as the number of observations increases this model does satisfy the invariance principle two sets of data are analysed in detail in the first example one red marble is observed in six drawings from a bag inferences from the imprecise dirichlet model are compared with objective bayesian and frequentist inferences the second example is an analysis of data from medical trials which compared two treatments for cardiorespiratory failure in newborn babies there are two problems to draw conclusions about which treatment is more effective and to decide when the randomized trials should be terminated this example shows how the imprecise dirichlet model can be used to analyse data in the form of a contingency table

a note on undominated lower probabilities on etudie la classe des probabilites inferieures non dominees on donne une construction simple et des extensions

factoriality of representations of the group of paths on su n abstract factoriality in the cyclic component of the vacuum for the energy representation of su n valued paths groups is proved the main tool is a lemma concerning generic pairs of cartan subalgebras in the lie algebra su n of su n groups

sub concave and sub convex capacities abstract in this study we propose the concepts of sub concave and sub convex capacities first we investigate some properties of sub concave and sub convex capacities second we discuss the choquet integrals with respect to sub concave and sub convex capacities we then show that the upper and lower probabilities comprising the supremum and infimum over the set of risk neutral martingale measures in the asymmetrical case are sub concave and sub convex capacities respectively

presentation and central extensions of mapping class groups we give a presentation of the mapping class groupm of a possibly bounded surface considering either all twists or just non separating twists as generators we also study certain central extensions of m one of them plays a key role in studying tqft functors namely the mapping class group of a p1 structure surface we give a presentation of this extension

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

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

law of large numbers for non additive measures our aim is to establish law of large numbers for classes of non additive measures for balanced games we obtain weak and strong law of large numbers for bounded random variables a sharper result is obtained for exact games we also provide an extension to upper envelope measures

robust bayesian analysis of selection models selection models arise when the data are selected to enter the sample only if they occur in a certain region of the sample space when this selection occurs according to some probability distribution the resulting model is often instead called a weighted distribution model in either case the original density becomes multiplied by a weight function w x often there is considerable uncertainty concerning this weight function for instance it may be known only that w lies between two specified weight functions we consider robust bayesian analysis for this situation finding the range of posterior quantities of interest such as the posterior mean or posterior probability of a set as w ranges over the class of weight functions the variational analysis utilizes concepts from variation diminishing transformations

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

strong law of large numbers with respect to a set valued probability measure in this paper we define the expected value of a random vector with respect to a set valued probability measure the concepts of independent and identically distributed random vectors are appropriately defined and a strong law of large numbers is derived in this setting finally an example of a set valued probability useful in bayesian inference is provided

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

stationary lower probabilities and unstable averages this paper explores the possibilities for probability like models of stationary nondeterministic phenomena that possess divergent but bounded time averages a random sequence described by a stationary probability measure must have almost surely convergent time averages whenever it has almost surely bounded time averages hence no measure can provide the mathematical model we desire in turning to lower probability based models we first explore the relationships between divergence stationarity and monotone continuity and those between monotone continuity and unicity of extensions we then construct several examples of stationary lower probabilities for sequences of uniformly bounded random variables such that divergence of time averages occurs with lower probability one we conclude with some remarks on the problem of estimating lower probability models on the basis of cylinder set observations

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

choquet expectation and peng s g expectation in this paper we consider two ways to generalize the mathematical expectation of a random variable the choquet expectation and peng s g expectation an open question has been after making suitable restrictions to the class of random variables acted on by the choquet expectation for what class of expectation do these two definitions coincide in this paper we provide a necessary and sufficient condition which proves that the only expectation which lies in both classes is the traditional linear expectation this settles another open question about whether choquet expectation may be used to obtain monte carlo like solution of nonlinear pde it cannot except for some very special cases

factorial representations of path groups abstract the reduction of the energy representation of the group of mappings from i 0 1 s11 r or r into a compact semisimple lie group g is given for g su 2 the factoriality of the representation which is of type iii in the case i r is proved

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

local sensitivity diagnostics for bayesian inference we investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k dimensional parameter space we show that several previously suggested diagnostics such as the norm of the frechet derivative diverge at rate n k 2 if the base prior is an interior point in the class of priors under the density ratio topology diagnostics based on divergences exhibit similar asymptotic behavior we show that better asymptotic behavior can be obtained by suitably restricting the classes of priors we also extend the diagnostics to see how various marginals of the prior affect various marginals of the posterior

bayesian inference with specified prior marginals abstract we show how to find bounds on posterior expectations of arbitrary functions of the parameters when the prior marginals are specified but when the complete joint prior is unspecified we also give a theorem that is useful for finding posterior bounds in a wide range of bayesian robustness problems we apply these techniques to two examples the first example involves a recent clinical trial for extracorporeal membrane oxygenation ecmo our analysis may be regarded as a follow up to a detailed bayesian analysis given by kass and greenhouse who concluded that the posterior probability that the treatment is superior to the control is about 95 their analysis however assumed a priori independence of the parameters we consider other prior distributions with the same marginals as kass and greenhouse but in which the parameters are not independent and conclude that as long as a priori independence is at least approximately tenable then ecmo seems superior to the control the second example is th

irreducibility and reducibility for the energy representation of the group of mappings of a riemannian manifold into a compact semisimple lie group abstract the irreducibility of the energy representation of the group of smooth mappings from a riemannian manifold of dimension d 3 into a compact semisimple lie group is proven the nonequivalence of the representations associated with different riemann structures is also proven for d 3 the case d 2 is examined and irreducibility and nonequivalence results are also given the reducibility in the case d 1 is pointed out in this case the commutant contains a representation equivalent with the energy representation

ranges of posterior probabilities for quasiunimodal priors with specified quantiles abstract suppose several quantiles of the prior distribution for are specified or equivalently the prior probabilities of a partitioning collection of intervals ii are given in addition suppose that the prior distribution is assumed to be unimodal rather than selecting a single prior distribution to perform a bayesian analysis it is of interest to consider the class of all prior distributions compatible with these inputs for this class and unimodal likelihood functions the ranges of the posterior probabilities of the ii and the ranges of the posterior cdf at the specified prior quantiles were determined in berger and o hagan in press unfortunately calculations with this class can be difficult here a similar much more easily analyzed class of quasiunimodal prior distributions is considered and compared with other classes

signature of relations in mapping class groups and non holomorphic lefschetz fibrations we introduce the notion of signature for relations in mapping class groups and show that the signature of a lefschetz fibration over the 2 sphere is the sum of the signatures for basic relations contained in its monodromy combining explicit calculations of the signature cocycle with a technique of substituting positive relations we give some new examples of non holomorphic lefschetz fibrations of genus 3 4 and 5 which violate slope bounds for non hyperelliptic fibrations on algebraic surfaces of general type

sensitivity in bayesian statistics the prior and the likelihood abstract one paradigm for sensitivity analyses in bayesian statistics is to specify a reasonable class of priors and to compute the corresponding class of posterior inferences the class is chosen to represent uncertainty about the prior there is often additional uncertainty however about the family of sampling distributions this article introduces a method for computing ranges of posterior expectations over reasonable classes of sampling distributions that lie close to a given parametric family by treating the prior as a probability measure on the space of sampling distributions this article also gives a unified treatment to what are usually considered two separate problems sensitivity to the prior and sensitivity to the sampling model first the notion of close to is made explicit then an algorithm is given for turning ratio linear problems into sequences of linear problems in addition to solving the problem at hand the algorithm simplifies many other robust bayesian computational pro

local sensitivity of inferences to prior marginals abstract the sensitivity of posterior expectations to perturbations of prior marginals is investigated using a local method this approach is particularly well suited to high dimensional parameter spaces qualitative and quantitative results are obtained regarding the propagation of sensitivity in multiparameter settings more specifically a posterior expectation is treated as a functional of the prior marginal the derivative of this functional is evaluated at the nominal prior marginal yielding a measure of sensitivity this is computationally more feasible than evaluating the range of a posterior quantity over a class of priors hence sensitivity can be assessed in high dimensional problems the local technique is used to assess the influence of distributional assumptions at the various stages of a hierarchical model

the extent of non conglomerability of finitely additive probabilities an arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive pfa one the coefficient of the pfa probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition if the probability is defined on a field the bound is sharp hence non conglomerability or equivalently non disintegrability characterizes finitely as opposed to countably additive probability nonetheless there exists a pfa probability which is simultaneously conglomerable over an arbitrary finite set of partitions neither conglomerability nor non conglomerability in a given partition is closed under convex combinations but the convex combination of pfa ultrafilter probabilities each of which cannot be made conglomerable in a common margin is singular with respect to any finitely additive probability that is conglomerable in that margin

local sensitivity of posterior expectations we investigate the degree to which posterior expectations are sensitive to prior distributions using a local method based on functional differentiation invariance considerations suggest a family of norms which can be used to measure perturbations to the prior the sensitivity measure is seen to depend heavily on the choice of norm asymptotic results suggest which norm will yield the most useful results in practice we find that to maintain asymptotically sensible behaviour it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases jeffreys prior is characterized as the prior to which inference is least sensitive

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

bayes like decision making with upper and lower probabilities abstract we consider the use of interval valued probabilities to represent the support lent to the hypothesis that the parameter value lies in a subset a of the parameter set when we observe x know the likelihoods f e and have some prior information concerning the parameter our model for prior information is that of a salient prior distribution in which we have little confidence although we have much less confidence in any alternative prior we consider notions of acceptable and coherent decision making as well as notions of being able to achieve a bayes rule and least commitment throughout we are motivated to preserve some of the elements of bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge

imprecise data sets as a source of ambiguity a model and experimental evidence in many circumstances evaluations are based on empirical data however some observations may be imprecise meaning that it is not entirely clear what occurred in them we address the question of how beliefs are formed in these situations the individual in our model is essentially a frequentist he first makes a subjective judgment about the occurrence of the event for each imprecise observation this may be any number between zero and one he then evaluates the event by its subjective frequency of occurrence our model connects the method of processing imprecise observations with the individual s attitude toward ambiguity an individual who in imprecise observations puts low high weight on the possibility that an event occurred is ambiguity averse loving an experiment supports the main assertions of the model with precise data subjects behave as if there were no ambiguity whereas with imprecise data subjects turn out to be ambiguity averse this paper was accepted by brad barber teck ho and terrance odean special issue editors

ranges of posterior measures for priors with unimodal contaminations we consider the problem of robustness or sensitivity of given bayesian posterior criteria to specification of the prior distribution criteria considered include the posterior mean variance and probability of a set for credible regions and hypothesis testing uncertainty in an elicited prior go is modelled by an e contamination class f x 1 c 1t0 eq q e q where reflects the amount of probabilistic uncertainty in g and q is a class

invariance properties of density ratio priors density ratio neighborhoods are classes of probabilities that are used in robust bayesian inference these classes are invariant under bayesian updating and marginalization this makes them computationally convenient in robust bayesian inference we show that this is the unique class of probabilities that has these invariance properties aside from its theoretical value this result has computational implications as well

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

robust bayesian experimental designs in normal linear models we address the problem of finding a design that minimizes the bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior uncertainty in the prior is formulated in terms of having a family of priors instead of one single prior two different classes of priors are considered f1 is a family of conjugate priors and a second family of priors f2 is induced by a metric on the space of nonnegative measures the family f1 has earlier been suggested by leamer and polasek while f2 was considered by derobertis and hartigan and berger the setup assumed is that of a canonical normal linear model with independent homoscedastic errors optimal robust designs are considered for the problem of estimating the vector of regression coefficients or a linear combination of the regression coefficients and also for testing and set estimation problems concrete examples are given for polynomial regression and completely randomized designs a very surprising finding is that for r2 the same design is optimal for a variety of different problems with different loss structures in general the results for f2 are significantly more substantive our results are applicable to group decision making and reconciliation of opinions among experts with different priors

symplectic surgeries along certain singularities and new lefschetz fibrations abstract we define a new 4 dimensional symplectic cut and paste operations arising from the generalized star relations t a 0 t a 1 t a 2 t a 2 g 1 2 g 1 t b 1 t b 2 g t b 3 also known as the trident relations in the mapping class group g 3 of an orientable surface of genus g 1 with 3 boundary components we also construct new families of lefschetz fibrations by applying the generalized star relations and the chain relations to the families of words t c 1 t c 2 t c 2 g 1 t c 2 g t c 2 g 1 2 t c 2 g t c 2 g 1 t c 2 t c 1 2 n 1 t c 1 t c 2 t c 2 g t c 2 g 1 2 g 2 n 1 and t c 1 t c 2 t c 2 g 1 t c 2 g 2 2 g 1 n 1 in the mapping class group g of the closed orientable surface of genus g 1 and n 1 furthermore we show that the total spaces of some of these lefschetz fibrations are irreducible exotic symplectic 4 manifolds using the degenerate cases of the generalized star relations we also realize all elliptic lefschetz fibrations and genus two lefschetz fibrations over s 2 with non separating vanishing cycles

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

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

simultaneously least favorable experiments a concept of worst case sufficiency is defined generalizing le cam s approximate sufficiency instead of using total variation norm as did le cam 1964 neighborhoods are described by upper expectations a corresponding version of the theorem of le cam blackwell sherman stein is proved in the case of finite parameter space as a main tool serve standard experiments and their upper limits here to be called upper standard functionals a characterization of simultaneously least favorable experiments dominated by a family of upper expectations is proved it says that least favorable experiments exist if and only if the upper standard functional acts additively on a cone of concave functions

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

continuous lower probability based models for stationary processes with bounded and divergent time averages on developpe un nouveau type de modele stochastique pour des processus empiriques non deterministes qui exhibent des caracteristiques paradoxales de stationarite de variables bornees et de moyennes temporelles instables

a robust bayesian interpretation of likelihood regions likelihood regions are shown to be robust in the sense that their posterior probability content is relatively insensitive to contaminations of the prior this provides a bayesian interpretation of regions that are commonly used by frequentists to construct confidence intervals and whose use are also advocated by the pure likelihood approach

small lefschetz fibrations and exotic 4 manifolds we explicitly construct genus 2 lefschetz fibrations whose total spaces are minimal symplectic 4 manifolds homeomorphic to complex rational surfaces mathbb cp 2 p overline mathbb cp 2 for p 7 8 9 and to 3 mathbb cp 2 q overline mathbb cp 2 for q 12 ldots 19 complementarily we prove that there are no minimal genus 2 lefschetz fibrations whose total spaces are homeomorphic to any other simply connected 4 manifold with b le 3 with one possible exception when b 3 meanwhile we produce positive dehn twist factorizations for several new genus 2 lefschetz fibrations with small number of critical points including the smallest possible example which follow from a reverse engineering procedure we introduce for this setting we also derive exotic minimal symplectic 4 manifolds in the homeomorphism classes of mathbb cp 2 4 overline mathbb cp 2 and 3 mathbb cp 2 6 overline mathbb cp 2 from small lefschetz fibrations over surfaces of higher genera

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

decision making with monotone lower probabilities of infinite order fabrice properties of convex and monotone capacities of infinite order in polish spaces are studied and used to justify the representation of certain situations of imprecise risk imprecisely known probabilities by lower probabilities which are monotone of infinite order decision making with imprecise risk is then modeled and linear utility theory is shown to be generalizable to the case of imprecise risk

network of probabilities associated with a capacity of order 2 abstract this paper studies capacities of order 2 from a geometric point of view that is from convex polytope of probabilities defined by them this convex polytope is characterized by means of a network of probabilities representing relationships among vertices defined by the edges interesting properties may be obtained about the capacities from this network

an approach to robust bayesian analysis for multidimensional parameter spaces abstract in some multiparameter bayesian problems the prior is more naturally described as a joint density for all of the parameters and in others it is more naturally described as a product of marginal and conditional densities although ordinary quantile e contamination and density ratio classes of priors may be sensible choices for robust bayesian analyses in the former case they may be less satisfactory in the latter this article shows how to construct product classes of priors for use in the second case using one class of priors for the marginal and another for each conditional then the article introduces density bounded classes as an alternative to the more familiar classes for representing uncertainty about the prior it is often of interest to find tight bounds on posterior expectations as the prior ranges over the class of priors an apparently difficult nonlinear optimization problem regardless of which class of priors is used this article describes a technique for turning the problem into

extensions of fuzzy aggregation abstract we discuss the equivalence between aggregation of fuzzy sets and integration with respect to a special class of non additive set functions both fuzzy integral and choquet integral are considered first we study aggregation of a finite family of fuzzy sets and then we extend our results to aggregation of an infinite family the concepts of comonotonic maxitivity and additivity play a central role we argue that for the purpose of aggregating fuzzy sets comonotonic maxitivity is a more desirable requirement than comonotonic additivity in the absence of any such requirement we explore a wider class of aggregation procedures another important subject that we study is a comparison between two aggregation functions and we obtain a simple characterization in both the finite and infinite case our results are related to concepts that were studied in the three areas where non additive set functions and integrals were found particularly useful fuzzy sets theory mathematical economics and mathematical statistics

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

asymptotic global robustness in bayesian decision theory in bayesian decision theory it is known that robustness with respect to the loss and the prior can be improved by adding new observations in this article we study the rate of robustness improvement with respect to the number of observations n three usual measures of posterior global robustness are considered the range of the bayes actions set derived from a class of loss functions the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class we show that the rate of convergence of the first measure of robustness is n while it is n for the other measures under reasonable assumptions on the class of loss functions we begin with the study of two particular cases to illustrate our results

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

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subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

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

meyer s signature cocycle and hyperelliptic fibrations we show that the cohomology class represented by meyer s signature cocycle is of order 2g 1 in the 2 dimensional cohomology group of the hyperelliptic mapping class group of genus g by using the 1 cochain cobounding the signature cocycle we extend the local signature for singular fibers of genus 2 fibrations due to y matsumoto 18 to that for singular fibers of hyperelliptic fibrations of arbitrary genus g and calculate its values on lefschetz singular fibers finally we compare our local signature with another local signature which arises from algebraic geometry

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

on the huber strassen theorem we show that the huber strassen theorem can be applied to many common robust neighborhood models without restrictive compactness assumptions our method is to replace the original polish topology with a compact topology which is possible according to the kuratowski isomorphism theorem