Extrapolating Validation Using Bayesian Networks with Interval Probabilities

The potential applications and challenges of uncertain data mining have recently attracted interests from researchers. Most uncertain data mining algorithms consider aleatory (random) uncertainty of data, i.e. these algorithms require that exact probability distributions or confidence values are attached to uncertain data. However, knowledge about uncertainty may be incomplete in the case of epistemic (incomplete) uncertainty of data, i.e. probabilities of uncertain data may be imprecise, coarse, or missing in some applications. The paper focuses on uncertain data which miss probabilities, specially, value-uncertain discrete objects which miss probabilities (for short uncertain objects). On the other hand, classification is one of the most important tasks in data mining. But, to the best of our knowledge, there is no method to learn Naïve Bayesian classifier from uncertain objects. So the paper studies Naïve Bayesian classification of uncertain objects. Firstly, the paper defines interval probabilities of uncertain objects from probabilistic cardinality point of view, and bridges the gap between uncertain objects and the theory of interval probability by proving that interval probabilities are F-probabilities. Secondly, based on the theory of interval probability, the paper defines conditional interval probabilities including the intuitive concept and the canonical concept, and the conditional independence of the intuitive concept. Further, the paper gives a formula to effectively compute the intuitive concept. Thirdly, the paper presents a Naïve Bayesian classifier with interval probability parameters which can handle both uncertain objects and certain objects. Finally, experiments with uncertain objects based on UCI data show satisfactory performances.

Bayesian networks are often proposed as a method for high-level information fusion. However, a Bayesian network relies on strong assumptions about the underlying probabilities. In many cases it is not realistic to require such precise probability assessments. We show that there exists a significant set of problems where credal networks outperform Bayesian networks, thus enabling more dependable decision making for this type of problems. A credal network is a graphical probabilistic method that utilizes sets of probability distributions, e.g., interval probabilities, for representation of belief. Such a representation allows one to properly express epistemic uncertainty, i.e., uncertainty that can be reduced if more information becomes available. Since reducing uncertainty has been proposed as one of the main goals of information fusion, the ability to represent epistemic uncertainty becomes an important aspect in all fusion applications.

As a dominant method in evidential reasoning, Bayesian network has been proved powerful in discrete fields. Although Bayesian network performs reliable in continuous variables and interval estimations, it relies on discretizing continuous variables or building an approximate model to conduct, which causes information loss and accuracy reduction. In order to bridge this gap, this paper introduces two inference rules combined with four inference rules proposed by other scholars. Then we propose a concept of uncertain inference network that consists of six basic structures matching inference rules to represent relationships and logic connection among the evidence. Evidence is represented by uncertain sets that can apply to continuous variables using membership functions to represent vague concepts. Furthermore, a numeric experiment for a forensic investigation of fire incidents is given to compare the results of uncertain inference network and Bayesian network. We found three merits in the case study. First, an uncertain inference network has simpler data access for each node because Bayesian network depends on conditional probability tables while uncertain inference network only relies on membership function. Second, an uncertain inference network has a more wide application because it can perform continuous variables with certain mathematical formulas without discretizing or approximating. Third, an uncertain inference network has a more accurate result because Bayesian network gives a point estimation with a 0–1 value while uncertain inference network conducts an interval estimation with a range value.

Interval data are widely used in real applications to represent the values of quantities in uncertain situations. However, the implied probabilistic causal relationships among interval-valued variables with interval data cannot be represented and inferred by general Bayesian networks with point-based probability parameters. Thus, it is desired to extend the general Bayesian network with effective mechanisms of representation, learning and inference of probabilistic causal relationships implied in interval data. In this paper, we define the interval probabilities, the bound-limited weak conditional interval probabilities and the probabilistic description, as well as the multiplication rules. Furthermore, we propose the method for learning the Bayesian network structure from interval data and the algorithm for corresponding approximate inferences. Experimental results show that our methods are feasible, and we conclude that the Bayesian network with interval probability parameters is the expansion of the general Bayesian network.

During the last decades, "Transport Demand" and "Mobility" has been a continuously developing branch in the transport literature.This is reflected in the great amount of research papers published in scientific magazines dealing with trip matrix estimation (see (Doblas & Benítez, 2005)) and traffic assignment problems (see (Praskher & Bekhor, 2004)).Current traffic models reproduce the mobility using several data inputs, in particular prior trip matrix, link counts, etc. (see (Yang & Zhou, 1998)) which are data only from a subset of the problem variables, and its size will depend on the available budget for the study being carried on.Among the problems faced for solving these models we can emphasize the high number of possible solutions which is usually solved by choosing the solution where the model results best fits real data.Nevertheless a model does not have to reproduce only the real data, but also must reproduce accurately all the variables.To this end, the aim of this paper consists of presenting two Bayesian network models for traffic estimation, trying to bring a new tool to the transportation field.The first one deals with the problem of link flows, trip matrix estimation and traffic counting location and in the second one we propose a Bayesian network for route flow estimation (and hence link flows and OD flows) using data from plate scanning technique together with a model for optimal plate scanning device location.Since a Gaussian Bayesian network is used, these models allow us to update the predictions from a small subset of real data and probability intervals or regions are obtained to get an idea of the associated uncertainties.In addition dealing with data from the plate scanning approach we improve the under-specification level of the traffic flow estimation problem. Some background on Bayesian network modelsA Bayesian network (see, (Castillo et al., 1999)) is a pair (G, P ), where G is a directed acyclic graph (DAG) defined on a set of nodes X (the random variables), and P = {p(x 1 |π 1 ), . . ., p(x n |π n )} is a set of n conditional probability densities (CPD), one for each variable, and Π i is the set of parents of node X i in G.The set P defines the associated joint 19 www.intechopen.comBayesian Network 318 probability density of all nodes as p(1)The graph G contains all the qualitative information about the relationships among the variables, no matter which probability values are assigned to them 1 .Complementary, the probabilities in P contain the quantitative information, i.e., they complement the qualitative properties revealed by the graphical structure.

This paper proposes machine learning techniques to discover knowledge in a dataset in the form of if-then rules for the purpose of formulating queries for validation of a Bayesian belief network model of the same data. Although do-main expertise is often available, the query formulation task is tedious and laborious, and hence automation of query formulation is desirable. In an effort to automate the query formulation process, a machine learning algorithm is lev-eraged to discover knowledge in the form of if-then rules in the data from which the Bayesian belief network model under validation was also induced. The set of if-then rules are processed and filtered through domain expertise to identify a subset that consists of “interesting” and “significant” rules. The subset of interesting and significant rules is formulated into corresponding queries to be posed, for validation purposes, to the Bayesian belief network induced from the same dataset. The promise of the proposed methodology was assessed through an empirical study performed on a real-life dataset, the National Crime Victimization Survey, which has over 250 attributes and well over 200,000 data points. The study demonstrated that the proposed approach is feasible and provides automation, in part, of the query formulation process for validation of a complex probabilistic model, which culminates in substantial savings for the need for human expert involvement and investment.

The perhaps most important signaling network in living cells is constituted by the interactions of proteins with the genome – the regulatory genetic network of the cell. From a system-level point of view, the various interactions and control loops, which form a genetic network, represent the basis upon which the vast complexity and flexibility of life processes emerges. Here we provide a review over some efforts towards gaining a quantitative understanding of regulatory genetic networks by means of large scale computational models. After a brief description of the biological principles of gene regulation, we summarize recent advances in massive gene-expression measurements by DNA-microarrays, which form the to date most powerful data basis for models of genetic networks. One class of models such as reaction-diffusion networks and nonlinear dynamical descriptions are biased towards using explicit molecular biological knowledge. A second class, centered around machine learning approaches like neural networks and Bayesian networks, adopts a more data-driven approach and thereby makes massive use of the novel gene expression measurement techniques.

ABSTRACTThe hierarchical validation is recommended for assessing the confidence of the large-scale simulation model, through comparing the model predictions with experimental observations at each level. However, the system-level experimental observations may be scarce in the full-scale testing. A new structural modelling approach is proposed to implement the higher-level model validation using lower-level data. The proposed approach relates the computation model to corresponding system responses and lower-level outputs to higher-level responses. Bayesian network with Markov chain Monte Carlo (MCMC) sampling is used to represent the relationship sets and estimate the structural modelling parameters and input variables. Highest posterior density (HPD) confidence range is employed to quantify the model credibility, the confidence intervals of model errors, and the effect of lower-level data on the higher-level model assessment. The proposed methodology is implemented for hierarchical model validation and confidence extrapolation of a guidance model on a flight vehicle using time series data.

We propose a Bayesian higher-order probability logic reasoning approach with interval probability parameters to the problem of making inference from conditional knowledge, which combines weak conditional probability and conditional event algebra for approximate inferences. We define the bound-limited weak conditional interval probabilities, the corresponding probabilistic description, and the multiplication rules of weak conditional probabilities for joint probability distribution, and use higher-order conditional event to resolve a discrepancy between logic and probability. By extending normal measurable space with conditional event, we bring logic consistent with probability in denoting conditional knowledge, and then transform a higher-order conditional event to normal events and corresponding logical joint events via conditional event algebra. Based on multiplication rules, we compute the quantitative values of the events with interval parameters, and evaluate the value of higher-order conditional event and finish reasoning process. An illustrative example of application of our method shows how we make inferences from conditional knowledge.

One of the major challenges for product lifecycle management systems is the lack of integrated decision support tools to help decision-making with available information in collaborative enterprise networks. Uncertainties are inherent in such networks due to lack of perfect knowledge or conflicting information. In this paper, a robust decision support approach based on imprecise probabilities is proposed. Robust Bayesian belief networks with interval probabilities are used to estimate imprecise posterior probabilities in probabilistic inference. This generic approach is demonstrated with decision-makings in design for closed-loop supply chain. The ultimate goal of robust intelligent decision support systems is to enhance the effective use of information available in collaborative engineering environments.

A random set approach to the optimization of uncertain structures

A new fast static security analysis algorithm, based on the interval analysis theory and probability theory, is proposed to improve the efficiency and accuracy of security analysis during power network interaction. In this paper, the characteristics of network interaction and uncertainty in the interaction process are illustrated firstly. After that the dimension reduction technique is employed in DC interval load flow calculation to determine the branch power flow efficiently. Then active power related contingency screening index is proposed, based on the DC interval analysis and the probability theory, which facilitates the screening of serious contingencies during the interactive process and improves the accuracy of contingency screening. Finally, cumulated and Gram-Charlier series expansion based AC probabilistic power flow is carried out for different operating scenarios during interactive time process. JS 1018-bus example system is introduced to validate the proposed method. Compared with traditional static security analysis, the proposed method can provide more efficient and accurate results for real time system operation.

On the bounds for structural system reliability

Reliability of corrosion-damaged steel structures using interval probability theory

Survival prediction for gallbladder carcinoma after curative resection: Comparison of nomogram and Bayesian network models