Probability intervals over influence diagrams

A mechanism for performing probabilistic reasoning in influence diagrams using interval rather than point-valued probabilities is described. Procedures for operations corresponding to conditional expectation and Bayesian conditioning in influence diagrams are derived where lower bounds on probabilities are stored at each node. The resulting bounds for the transformed diagram are shown to be the tightest possible within the class of constraints on probability distributions that can be expressed exclusively as lower bounds on the component probabilities of the diagram. Sequences of these operations can be performed to answer probabilistic queries with indeterminacies in the input and for performing sensitivity analysis on an influence diagram. The storage requirements and computational complexity of this approach are comparable to those for point-valued probabilistic inference mechanisms     

基于可达概率区间的不确定决策树

针对不确定数据的概率分布难以获取的客观实际,讨论了缺失概率分布的值不确定离散对象的决策树。定义了(条件)概率区间,并证明了(条件)概率区间是可达概率区间;基于可达概率区间,定义了(条件)熵区间,并给出了求解(条件)熵区间的上/下界的方法;采用条件熵区间作为属性选择度量,提出了一种新的不确定决策树,将以0-1划分对象的决策树扩展到以概率区间分配对象的决策树,这样不仅可以处理缺失概率分布的值不确定离散对象,也可以处理确定离散对象。通过在基于UCI数据集的不确定数据集上的实验,证实了不确定决策树是有效的。     

Robust reasoning with rules that have exceptions: From second-order probability to argumentation via upper envelopes of probability and possibility plus directed graphs

Rules having rare exceptions may be interpreted as assertions of high conditional probability. In other words, a rule If X then Y may be interpreted as meaning that Pr(Y∣X) 1. A general approach to reasoning with such rules, based on second-order probability, is advocated. Within this general approach, different reasoning methods are needed, with the selection of a specific method being dependent upon what knowledge is available about the relative sizes, across rules, of upper bounds on each rule's exception probabilities Pr(?Y∣X). A method of reasoning, entailment with universal near surety, is formulated for the case when no information is available concerning the relative sizes of upper bounds on exception probabilities. Any conclusion attained under these conditions is robust in the sense that it will still be attained if information about the relative sizes of exception probability bounds becomes available. It is shown that reasoning via entailment with universal near surety is equivalent to reasoning in a particular type of argumentation system having the property that when two subsets of the rule base conflict with each other, the effectively more specific subset overrides the other. As stepping stones toward attaining this argumentation result, theorems are proved characterizing entailment with universal near surety in terms of upper envelopes of probability measures, upper envelopes of possibility measures, and directed graphs. In addition, various attributes of entailment with universal near surety, including property inheritance, are examined.     

Unified uncertainty analysis by the mean value first order saddlepoint approximation

Uncertainties exist in products or systems widely. In general, uncertainties are classified as epistemic uncertainty or aleatory uncertainty. This paper proposes a unified uncertainty analysis (UUA) method based on the mean value first order saddlepoint approximation (MVFOSPA), denoted as MVFOSPA-UUA, to estimate the systems probabilities of failure considering both epistemic and aleatory uncertainties simultaneously. In this method, the input parameters with epistemic uncertainty are modeled using interval variables while input parameters with aleatory uncertainty are modeled using probability distribution or random variables. In order to calculate the lower and upper bounds of system probabilities of failure, both the best case and the worst case scenarios of the system performance function need to be considered, and the proposed MVFOSPA-UUA method can handle these two cases easily. The proposed method is demonstrated to be more efficient, robust and in some situations more accurate than the existing methods such as uncertainty analysis based on the first order reliability method. The proposed method is demonstrated using several examples.     

Upper bounds on the connection probability for 2-D meshes and tori

Mesh is an important and popular interconnection network topology for large parallel computer systems. A mesh can be divided into submeshes to obtain the upper bounds on the connection probability for the mesh. Combinatorial techniques are used to get closer upper bounds on the connection probability for 2-D meshes compared with the existing upper bounds we have known. Simulation results of meshes of various sizes show that our upper bounds are close to the exact connection probability. The combinatorial methods and tools used in this paper can be used to study the connection probabilities for other networks.     

Misclassification Probability Bounds for Multivariate Gaussian Classes

Classification systems based on linear discriminant analysis are employed in a variety of communications applications, in which the classes are most commonly characterized by known Gaussian PDFs. The performance of these classifiers is analyzed in this paper in terms of the conditional probability of misclassification. Easily computed lower and upper bounds on this error probability are presented and shown to provide corresponding bounds on the number of Monte Carlo trials required to obtain a desired level of accuracy. The error probability bounds yield an exact and easily computed expression for the error probability in the case where there are only two classes and a single hyperplane. In the special case where misclassification into a nominated class is independent of all other misclassifications, successively tighter upper and lower bounds can be computed at the expense of successively higher-order products of the individual misclassification probabilities. Finally, bounds are provided on the number of Monte Carlo trials required to improve, with suitably high confidence level, on the confidence interval formed by the error probability bounds.     

Credal网络推理的一种不完全枚举法

Credal网络是研究不确定环境下知识表示和因果推理的一种图模型,其条件概率值可以用不精确的区间或不等式定性地表示,使得表达方式更加灵活有效。Credal网络的推理是计算一定证据下的后验概率最大值和最小值,给出了一种Credal网络推理的新方法,该方法是在桶消元框架下通过枚举计算部分因子函数值,使计算量大大减小,并且可以得到精确的结果。最后用一个实例说明了该方法的可行性。     

Sensitivity analysis in discrete Bayesian networks

This paper presents an efficient computational method for performing sensitivity analysis in discrete Bayesian networks. The method exploits the structure of conditional probabilities of a target node given the evidence. First, the set of parameters which is relevant to the calculation of the conditional probabilities of the target node is identified. Next, this set is reduced by removing those combinations of the parameters which either contradict the available evidence or are incompatible. Finally, using the canonical components associated with the resulting subset of parameters, the desired conditional probabilities are obtained. In this way, an important saving in the calculations is achieved. The proposed method can also be used to compute exact upper and lower bounds for the conditional probabilities, hence a sensitivity analysis can be easily performed. Examples are used to illustrate the proposed methodology     

Automated reasoning using possibilistic logic: semantics, beliefrevision, and variable certainty weights

An approach to automated deduction under uncertainty, based on possibilistic logic, is described; for that purpose we deal with clauses weighted by a degree that is a lower bound of a necessity or a possibility measure, according to the nature of the uncertainty. Two resolution rules are used for coping with the different situations, and the classical refutation method can be generalized with these rules. Also, the lower bounds are allowed to be functions of variables involved in the clauses, which results in hypothetical reasoning capabilities. In cases where only lower bounds of necessity measures are involved, a semantics is proposed in which the completeness of the extended resolution principle is proved. The relation between our approach and the idea of minimizing abnormality is briefly discussed. Moreover, deduction from a partially inconsistent knowledge base can be managed in this approach and captures a form of nonmonotonicity     

Stronger uncertainty relations with improvable upper and lower bounds

The quantum superposition principle is used to establish improved upper and lower bounds for the Maccone–Pati uncertainty inequality, which is based on a “weighted-like” sum of the variances of observables. Our bounds include free parameters that not only guarantee nontrivial bounds but also effectively control the bounds’ tightness. Significantly, these free parameters depend on neither the state nor the observables. A feature of our method is that any nontrivial bound can always be improved. In addition, we generalize both bounds to uncertainty relations with multiple (three or more) observables, maintaining the uncertainty relations’ tightness. Examples are given to illustrate our improved bounds.     

Uncertain inheritance and recognition as probabilistic default reasoning

This paper proposes probabilistic default reasoning as a suitable approach to uncertain inheritance and recognition for fuzzy and uncertain object-oriented models. The uncertainty is due to the uncertain membership of an object to a class and/or the uncertain applicability of a property, i.e., an attribute or a method, to a class. First, we introduce a logic-based uncertain object-oriented model where uncertain membership and applicability are measured by support pairs, which are lower and upper bounds on probability. The probability for a property being applicable to a class is interpreted as the conditional probability of the property being applicable to an object given that the object is a member of the class. Each uncertainty applicable property is then a default probabilistic logic rule, which is defeasible. In order to reduce the computational complexity of general probabilistic default reasoning, we propose to use Jeffrey's rule for a weaker notion of consistency and for local inference, then apply them to uncertain inheritance of attributes and methods. Using the same approach but with inverse Jeffrey's rule, uncertain recognition as probabilistic default reasoning is also presented. © 2001 John Wiley & Sons, Inc.     

Stochastic dynamics simulation with generalized interval probability

Two types of uncertainties are generally recognized in modelling and simulation, including variability caused by inherent randomness and incertitude due to the lack of perfect knowledge. In this paper, a generalized interval-probability theory is used to model both uncertainty components simultaneously, where epistemic uncertainty is quantified by generalized interval in addition to probability measure. Conditioning, independence, and Markovian probabilities are uniquely defined in generalized interval probability such that its probabilistic calculus resembles that in the classical probability theory. A path-integral approach can be taken to solve the interval Fokker–Planck equation for diffusion processes. A Krylov subspace projection method is proposed to solve the interval master equation for jump processes. Thus, the time evolution of both uncertainty components can be simulated simultaneously, which provides the lower and upper bound information of evolving probability distributions as an alternative to the traditional sensitivity analysis.     

System reliability using trivariate and bivariate integrals

Probabilistic safety analysis of civil engineering structures (system reliability analysis) requires the evaluation of multi-normal integrals and these multi-normal integrals can be estimated either from upper and lower bounds on these probabilities or by first-order second-moment approximate methods or by simulation methods. In this paper a new method based on conditional probabilities for the evaluation of series and parallel structural systems reliabilities is presented. Examples are studied to compare the accuracy of this new approach with other approximate methods.     

Probabilistic Reasoning Under Coherence in System P

In this paper we apply a probabilistic reasoning under coherence to System P. We consider a notion of strict probabilistic consistency, we show its equivalence to Adams' probabilistic consistency, and we give a necessary and sufficient condition for probabilistic entailment. We consider the inference rules of System P in the framework of coherent imprecise probabilistic assessments. Exploiting our coherence-based approach, we propagate the lower and upper probability bounds associated with the conditional assertions of a given knowledge base, obtaining the precise probability bounds for the derived conclusions of the inference rules. This allows a more flexible and realistic use of System P in default reasoning and provides an exact illustration of the degradation of the inference rules when interpreted in probabilistic terms. We also examine the disjunctive Weak Rational Monotony rule of System P⁺ proposed by Adams in his extended probabilistic logic. Finally, we examine the propagation of lower bounds with real -values and, to illustrate our probabilistic reasoning, we consider an example.     

Hybrid approaches for the two‐scenario max–min knapsack problem

In this paper, we deal with the two‐scenario max–min knapsack (MNK) problem. First, we consider several formulations of MNK as a mixed integer programming problem. Then, we propose a hybrid method as an alternative to solve the MNK exactly. The approach combines relaxation technique and the temporary setting of variables to improve iteratively two sequences of upper and lower bounds. More precisely, pseudo‐cuts are added to the problem to strengthen the bounds and reduce the gap between the best lower bound and the best upper bound. The algorithm stops when the proof of the optimality of the best solution is found. We also use a reduction technique to set some variables definitively at their optimal values. Numerical experiments demonstrate the robustness of the approach. In particular, our algorithm is efficient to solve large and correlated instances of MNK.     

基于小波变换的序列间距离严格估算

对时间序列的相似性搜索在很多新的数据库应用中的地位变得越来越重要．使用小波变换方法缩减维度是解决高维时间序列查询的一个有效方法．给出小波变换在时间序列相似性查找中对距离上下界的一个严格估计，同时说明传统的算法只是下界的一部分．根据给出的小波变换的下界，相对于传统的算法，可以排除更多的不相似序列．根据给出的上界，可以直接判断出两条序列是否相似，进一步减少需要验证的原始序列的个数．实验结果表明，相对于传统的算法，提出的上下界可以大幅度提高过滤效果，减少查询时间．     

Probabilistic guarantees for high-level robot behavior in the presence of sensor error

This paper presents an approach for reasoning about the effects of sensor error on high-level robot behavior. We consider robot controllers that are synthesized from high-level, temporal logic task specifications, such that the resulting robot behavior is guaranteed to satisfy these specifications when assuming perfect sensors and actuators. We relax the assumption of perfect sensing, and calculate the probability with which the controller satisfies a set of temporal logic specifications. We consider parametric representations, where the satisfaction probability is found as a function of the model parameters, and numerical representations, allowing for the analysis of large examples. We also consider models in which some parts of the environment and sensor have unknown transition probabilities, in which case we can determine upper and lower bounds for the probability. We illustrate our approach with two examples that provide insight into unintuitive effects of sensor error that can inform the specification design process.     

A review of: “KNOWLEDGE AS DESIGN”, by D. N. Perkins. Lawrence Earlbaum Associates,Hillsdale, New Jersey, 1986. XVI+ 247 pp.

The paper describes an approach to obtaining prior imprecise reliability values of components and systems. The mathematical basis of the approach is the theory of coherent imprecise probabilities. The procedure of prior imprecise probability elicitation is based on analogical reasoning, and two cases of precise and imprecise reliabilities of prototypes are considered. Cases of combining different reliability judgements for the same component are analyzed. The remainder of the paper is devoted to imprecise reliability assessments of different system structures: components in parallel, in series and in series-parallel. The formulae obtained allow getting the lower and upper probabilities without the assumption of conditional independence. What we can expect from the use of this new tool is briefly discussed.     

Nonlinear Filtering Under Stochastic Communication Protocol with Unknown Scheduling Probability

In this paper, the UKF-type nonlinear filtering problem is investigated for general nonlinear systems under stochastic communication protocols (SCPs) with unknown scheduling probabilities. In order to avoid the data collision and alleviate the network communication burden, SCPs, allowed only one sensor node to send data via the shared network, are exploited to orchestrate the scheduling order of sensor nodes. Different from traditional assumptions with accurate statistics, the scheduling probability of the selected node is unknown, but lies in a reliable interval with known upper and lower bounds. Due to the unknown probabilities, the exact estimation error covariance is not available and hence its upper bound is derived with the help of adding zero terms and eigenvalues of positive definite matrices. Such an upper bound is dependent on known upper and lower bounds of the scheduling probabilities and further utilized to reasonably design the filter gain at each time instant. In light of the obtained covariance and the filter gain, an improved unscented transformation is developed to carry out the designed UKF-type nonlinear filter by improving traditional approximate mean and covariance. Furthermore, the impact of the uncertain size of unknown scheduling probabilities is thoroughly discussed. Finally, a numerical example is given to confirm the effectiveness of the proposed nonlinear filter.