A theory of independent fuzzy probability for system reliability

Fuzzy fault trees provide a powerful and computationally efficient technique for developing fuzzy probabilities based on independent inputs. The probability of any event that can be described in terms of a sequence of independent unions, intersections, and complements may be calculated by a fuzzy fault tree. Unfortunately, fuzzy fault trees do not provide a complete theory: many events of substantial practical interest cannot be described only by independent operations. Thus, the standard fuzzy extension (based on fuzzy fault trees) is not complete since not all events are assigned a fuzzy probability. Other complete extensions have been proposed, but these extensions are not consistent with the calculations from fuzzy fault trees. We propose a new extension of crisp probability theory. Our model is based on n independent inputs, each with a fuzzy probability. The elements of our sample space describe exactly which of the n input events did and did not occur. Our extension is complete since a fuzzy probability is assigned to every subset of the sample space. Our extension is also consistent with all calculations that can be arranged as a fault tree. Our approach allows the reliability analyst to develop complete and consistent fuzzy reliability models from existing crisp reliability models. This allows a comprehensive analysis of the system. Computational algorithms are provided both to extend existing models and develop new models. The technique is demonstrated on a reliability model of a three-stage industrial process     

NAT model–based compression of Bayesian network CPTs over multivalued variables

Nonimpeding noisy‐AND tree (NAT) models offer a highly expressive approximate representation for significantly reducing the space of Bayesian networks (BNs). They also improve efficiency of BN inference significantly. To enable these advantages for general BNs, several technical advancements are made in this work to compress target BN conditional probability tables (CPTs) over multivalued variables into NAT models. We extend the semantics of NAT models beyond graded variables that causal independence models commonly adhered to and allow NAT modeling in nominal causal variables. We overcome the limitation of well‐defined pairwise causal interaction (PCI) bits and present a flexible PCI pattern extraction from target CPTs. We extend parameter estimation for binary NAT models to constrained gradient descent for compressing target CPTs over multivalued variables. We reveal challenges associated with persistent leaky causes and develop a novel framework for PCI pattern extraction when persistent leaky causes exist. The effectiveness of the CPT compression is validated experimentally.     

Learning symmetric causal independence models

Causal independence modelling is a well-known method for reducing the size of probability tables, simplifying the probabilistic inference and explaining the underlying mechanisms in Bayesian networks. Recently, a generalization of the widely-used noisy OR and noisy AND models, causal independence models based on symmetric Boolean functions, was proposed. In this paper, we study the problem of learning the parameters in these models, further referred to as symmetric causal independence models. We present a computationally efficient EM algorithm to learn parameters in symmetric causal independence models, where the computational scheme of the Poisson binomial distribution is used to compute the conditional probabilities in the E-step. We study computational complexity and convergence of the developed algorithm. The presented EM algorithm allows us to assess the practical usefulness of symmetric causal independence models. In the assessment, the models are applied to a classification task; they perform competitively with state-of-the-art classifiers.     

Efficient computation of marginal probabilities in multivalued causal inverted multiway trees given incomplete information

Maung and Paris [Internat J Intell Syst 1990, 5(5), 595–603] have shown that, in the general case, solving causal networks using maximum entropy techniques is NP complete. This paper considers multivalued causal inverted multiway trees, a nontrivial class of causal networks, in which any event can be influenced by any number of other events but itself only influences at most one event. We show that for this class of causal networks, maximum entropy can be used to find minimally prejudiced estimates for missing information. The techniques required for the current problem are substantially different from those used in the case of causal multiway trees in that nonlinear constraints arising from independence have to be incorporated. In addition, a new algebraic method is presented which isolates an unknown Lagrange multiplier by using the quotient of two pairs of state probabilities. Equating the joint probability distributions given by the Bayesian and maximum entropy models enables the Lagrange multipliers of the latter to be determined. An efficient iterative tree traversal algorithm which converges to the minimally prejudiced estimates for the missing information is described. When this information is added to that already provided, any existing method for updating the causal network can be used. ©1999 John Wiley & Sons, Inc.     

多预测子融合实时连续语音识别输出词正误判别

本文在采用堆栈译码词网重估输出作为识别最终输出的连续语音识别实时解码条件下,利用决策树方法将多个预测子融合,对识别输出词进行正确和错误的判别。本文首先构造了词后验概率、词长、相邻词的后验概率、词的声学和语言得分等共13 个预测子,然后利用决策树方法,通过选择不同的预测子组合方式和适当的决策树建树参数,筛选出预测子的最佳组合,建立优化的决策树进行输出词的正误判别。实验结果表明:利用局域词图计算的词后验概率与词长、相邻词的后验概率等几种实时预测子融合后,对识别输出词的正误判别能力得到提高,并且在实时性和分类效果两个方面优于n - best 输出的相应结果,相对于基线系统, 则分类错误率下降41. 4 %。实验结果也表明本文提出的相邻词的后验概率是相对重要的预测子。     

A method for evaluating elicitation schemes for probabilisticmodels

We present an objective approach for evaluating probability and structure elicitation methods in probabilistic models. The main idea is to use the model derived from the experts' experience rather than the true model as the standard to compare the elicited model. We describe a general procedure by which it is possible to capture the data corresponding to the expert's beliefs, and we present a simple experiment in which we utilize this technique to compare three methods for eliciting discrete probabilities: 1) direct numerical assessment, 2) the probability wheel, and 3) the scaled probability bar. We show that for our domain, the scaled probability bar is the most effective tool for probability elicitation.     

Recursive noisy OR - a rule for estimating complex probabilistic interactions

This paper focuses on approaches that address the intractability of knowledge acquisition of conditional probability tables in causal or Bayesian belief networks. We state a rule that we term the "recursive noisy OR" (RNOR) which allows combinations of dependent causes to be entered and later used for estimating the probability of an effect. In the development of this paper, we investigate the axiomatic correctness and semantic meaning of this rule and show that the recursive noisy OR is a generalization of the well-known noisy OR. We introduce the concept of positive causality and demonstrate its utility in axiomatic correctness of the RNOR. We also introduce concepts describing the ways in which dependent causes can work together as being either "synergistic" or "interfering." We provide a formalization to quantify these concepts and show that they are preserved by the RNOR. Finally, we present a method for the determination of Conditional Probability Tables from this causal theory.     

The efficient estimation of missing information in causal inverted multiway trees

Causal networks require probability values to be supplied for all possible combinations of outcomes in the cause–effect relationships implied by the network. Only then is it possible to use the existing methods for updating the information in the network to reflect new knowledge gained in a specific situation. Supplying causal information which is complete and accurate is not always possible in many applications, for example Decision Support Systems. This requirement becomes even more difficult to achieve when a single event is influenced by a large number of other events.Maximum Entropy can be used to find minimally prejudiced estimates for missing information but this approach is, in general, computationally infeasible.However, the authors have already shown that for certain special cases of causal networks such estimates can, in fact, be found in linear time.This article extends the work to causal inverted multiway trees in which any event can be influenced by any number of other events but itself only influences at most one event. In order to achieve this extension a thorough analysis of the traditional Bayesian model is undertaken to identify the large number of constraints which a valid Maximum Entropy model must satisfy. A simplified Maximum Entropy model is proposed and formal proofs that this satisfies the Bayesian properties are given.Equating the joint event probability distributions given by the Bayesian and Maximum Entropy models enables the Lagrange multipliers of the latter to be determined. This leads to an iterative tree traversal algorithm which converges to the minimally prejudiced estimates for the missing information. When this information is added to that already provided, any existing method for updating the causal network can be utilised.     

Interval-Valued Finite Markov Chains

The requirement that precise state and transition probabilities be available is often not realistic because of cost, technical difficulties or the uniqueness of the situation under study. Expert judgements, generic data, heterogeneous and partial information on the occurrences of events may be sources of the probability assessments. All this source information cannot produce precise probabilities of interest without having to introduce drastic assumptions often of quite an arbitrary nature. in this paper the theory of interval-valued coherent previsions is employed to generalise discrete Markov chains to interval-valued probabilities. A general procedure of interval-valued probability elicitation is analysed as well. In addition, examples are provided.     

Evidential reasoning using stochastic simulation of causal models

Stochastic simulation is a method of computing probabilities by recording the fraction of time that events occur in a random series of scenarios generated from some causal model. This paper presents an efficient, concurrent method of conducting the simulation which guarantees that all generated scenarios will be consistent with the observed data. It is shown that the simulation can be performed by purely local computations, involving products of parameters given with the initial specification of the model. Thus, the method proposed renders stochastic simulation a powerful technique of coherent inferencing, especially suited for tasks involving complex, nondecomposable models where “ballpark” estimates of probabilities will suffice.     

Santa Barbara microwave backscattering model for woodlands

The Santa Barbara microwave backscattering model for woodland vegetation with discontinuous tree canopies is described, with an emphasis on the construction of the model from probability-weighted sub-components. The modelling approach is to treat individual tree crowns as scatterers and attenuators, using the probabilities of scattering and attenuation to compute total backscatter. Four major model components are defined: surface backscattering, crown volume scattering, multi-path interactions between crown and ground, and double-bounce trunk-ground interactions. Each component is divided into subcomponents having distinct scattering and attenuation paths. The scattering of each subcomponent is computed and weighted by the probability of its occurrence. Total backscatter from a simulated woodland stand is computed by incoherent summation of the components. Recent revisions to the model have modified the subcomponent definitions and improved the probability formulation.     

基于数据挖掘技术的Web应用异常检测

本文提出的异常检测系统以Web日志文件作为输入,利用数据挖掘技术建立两种异常检测模型,分别对待测的Web请求记录输出五个异常概率,对各概率进行加权处理后得到一个最终的异常概率。     

Mechanisms for information elicitation

We study the computational aspects of information elicitation mechanisms in which a principal attempts to elicit the private information of other agents using a carefully selected payment scheme based on proper scoring rules. Scoring rules, like many other mechanisms set in a probabilistic environment, assume that all participating agents share some common belief about the underlying probability of events. In real-life situations however, the underlying distributions are not known precisely, and small differences in beliefs of agents about these distributions may alter their behavior under the prescribed mechanism.We examine two related models for the problem. The first model assumes that agents have a similar notion of the probabilities of events, and we show that this approach leads to efficient design algorithms that produce mechanisms which are robust to small changes in the beliefs of agents.In the second model we provide the designer with a more precise and discrete set of alternative beliefs that the seller of information may hold. We show that construction of an optimal mechanism in that case is a computationally hard problem, which is even hard to approximate up to any constant. For this model, we provide two very different exponential-time algorithms for the design problem that have different asymptotic running times. Each algorithm has a different set of cases for which it is most suitable. Finally, we examine elicitation mechanisms that elicit the confidence rating of the seller regarding its information.     

Causal independence for probability assessment and inference usingBayesian networks

A Bayesian network is a probabilistic representation for uncertain relationships, which has proven to be useful for modeling real-world problems. When there are many potential causes of a given effect, however, both probability assessment and inference using a Bayesian network can be difficult. In this paper, we describe causal independence, a collection of conditional independence assertions and functional relationships that are often appropriate to apply to the representation of the uncertain interactions between causes and effect. We show how the use of causal independence in a Bayesian network can greatly simplify probability assessment as well as probabilistic inference     

基于策略梯度算法的工作量证明中挖矿困境研究

针对区块链中工作量证明（PoW）共识机制下区块截留攻击导致的挖矿困境问题，将矿池间的博弈行为视作迭代的囚徒困境（IPD）模型，采用深度强化学习的策略梯度算法研究IPD的策略选择。利用该算法将每个矿池视为独立的智能体（Agent），将矿工的潜入率量化为强化学习中的行为分布，通过策略梯度算法中的策略网络对Agent的行为进行预测和优化，最大化矿工的人均收益，并通过模拟实验验证了策略梯度算法的有效性。实验发现，前期矿池处于相互攻击状态，平均收益小于1，出现了纳什均衡的问题；经过policy gradient算法的自我调整后，矿池由相互攻击转变为相互合作，每个矿池的潜入率趋于0，人均收益趋于1。实验结果表明，policy gradient算法可以解决挖矿困境的纳什均衡问题，最大化矿池人均收益。     

Learning first-order probabilistic models with combining rules

Many real-world domains exhibit rich relational structure and stochasticity and motivate the development of models that combine predicate logic with probabilities. These models describe probabilistic influences between attributes of objects that are related to each other through known domain relationships. To keep these models succinct, each such influence is considered independent of others, which is called the assumption of “independence of causal influences” (ICI). In this paper, we describe a language that consists of quantified conditional influence statements and captures most relational probabilistic models based on directed graphs. The influences due to different statements are combined using a set of combining rules such as Noisy-OR. We motivate and introduce multi-level combining rules, where the lower level rules combine the influences due to different ground instances of the same statement, and the upper level rules combine the influences due to different statements. We present algorithms and empirical results for parameter learning in the presence of such combining rules. Specifically, we derive and implement algorithms based on gradient descent and expectation maximization for different combining rules and evaluate them on synthetic data and on a real-world task. The results demonstrate that the algorithms are able to learn both the conditional probability distributions of the influence statements and the parameters of the combining rules.     

电子商务系统的数据挖掘与智能推荐预测的研究

本文对数据挖掘的概念做了简要的描述,并对决策树、关联规则和聚类三种数据挖掘算法做了分析比较,认为决策树算法虽然对于每一个项有更详细的模式且支持连续的输入,但不能扩展为大的目录;关联规则算法虽然快速、可伸缩,但是对算法的参数非常敏感;聚类算法虽然按相似性对数据进行分组,但是要设置复杂的参数和变量。     

Robust M-ary detection filters and smoothers for continuous-time jump Markov systems

In this paper, we consider a dynamic M-ary detection problem when Markov chains are observed through a Wiener process. These systems are fully specified by a candidate set of parameters, whose elements are, a rate matrix for the Markov chain and a parameter for the observation model. Further, we suppose these parameter sets can switch according to the state of an unobserved Markov chain and thereby produce an observation process generated by time varying (jump stochastic) parameter sets. Given such an observation process and a specified collection of models, we estimate the probabilities of each model parameter set explaining the observation. By defining a new augmented state process, then applying the method of reference probability, we compute matrix-valued dynamics, whose solutions estimate joint probabilities for all combinations of candidate model parameter sets and values taken by the indirectly observed state process. These matrix-valued dynamics satisfy a stochastic integral equation with a Wiener process integrator. Using the gauge transformation techniques introduced by Clark and a pointwise matrix product, we compute robust matrix-valued dynamics for the joint probabilities on the augmented state space. In these new dynamics, the observation Wiener process appears as a parameter matrix in a linear ordinary differential equation, rather than an integrator in a stochastic integral equation. It is shown that these robust dynamics, when discretised, enjoy a deterministic upper bound which ensures nonnegative probabilities for any observation sample path. In contrast, no such upper bounds can be computed for Taylor expansion approximations, such as the Euler-Maryauana and Milstein schemes. Finally, by exploiting a duality between causal and anticausal robust detector dynamics, we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations. A computer simulation demonstrating performance is included.     

Estimating parameters of forest patch transition models from gap models

An algorithm to estimate the parameter values of a transition forest landscape model (MOSAIC) from a gap model (FACET) is presented here. MOSAIC is semi-Markov; it includes random distributed holding times and fixed or deterministic delays in addition to transition probabilities. FACET is a terrain-sensitive version of ZELIG, a spatially explicit gap model. For each topographic class, the input to the algorithm consists of gap model tracer files identifying the cover type of each plot through time. These cover types or states are defined a priori. The method, based on individual plots of the FACET model, requires one FACET run initialized from the “gap” cover type and follows the time history of each plot. The algorithm estimates the transition probability by counting the number of transitions between each pair of states and estimates the fixed lags and the parameters of the probability density functions of the distributed delays by recording the times at which these transitions are made. These density functions are assumed to be Erlang; its two parameters, order and rate, are estimated using a nonlinear least squares procedure. Thus, as output, the algorithm produces four matrices at each terrain class: transition probabilities, fixed delays, and the two parameters for the Erlang distributions. The algorithm is illustrated by its application to two sites, high and low elevation, from the H.J. Andrews Forest in the Oregon Cascades. This scaling-up method helps to bridge the conceptual breach between landscape- and stand-scale models. To reflect landscape heterogeneity, the algorithm can be executed repetitively for many different terrain classes. While the method developed here focuses on FACET and MOSAIC, this general approach could be extended to use other fine-scale models or other forms of meta-models.     

Discovering causality in large databases

A causal rule between two variables, X M Y, captures the relationship that the presence of X causes the appearance of Y. Because of its usefulness (compared to association rules), techniques for mining causal rules are beginning to be developed. However, the effectiveness of existing methods (such as the LCD and CU-path algorithms) are limited to mining causal rules among simple variables, and are inadequate to discover and represent causal rules among multi-value variables. In this paper, we propose that the causality between variables X and Y be represented in the form X M Y with conditional probability matrix M Y|X . We also propose a new approach to discover causality in large databases based on partitioning. The approach partitions the items into item variables by decomposing "bad" item variables and composing "not-good" item variables. In particular, we establish a method to optimize causal rules that merges the "useless" information in conditional probability matrices of extracted causal rules.