Bayesian belief networks

A Bayesian (belief) network is a graphical model for probabilistic relationships among a set of variables whose joint probability distributions are compactly represented in relation to future data. The Bayesian network has several advantages in modeling biological network systems  

2 Conditional and Joint Pmbability Distributions In addition to its graphical structure, a Bayesian network needs to be speerfied by the conditional probability distribution of eaeh node given its parents. Let A and D be variables of interest with a direct causal (parental) relationship in Example 11.6. This relationship can be represented by a conditional probabUity distribution P D A) which represents the probabilistic distribution of child node D given the information of parent node A. When both child and parent nodes arc discrete variables, a contingeney table can summarize the conditional probabdities for aU possible states given each of its parent node states. For continuous variables, a eonditional probability density function needs to be defined. For the combination of continuous and discrete nodes, a mixture distribution, for example, mixture normal distribution, will be required (Imoto et al., 2002). 

The joint probabUity distribution of a Bayesian network has aU the information about the relationships between variables in the network. However, as the number of nodes increases, the complexity of the joint probabUity distribution also increases rapidly. Fortunately, there is a simpler way to represent the joint probability distribution of a Bayesian network with fewer parameters. The so-called Markov conditional independence properties of the Bayesian network can be used to decompose the whole DAG S into subcomponents of directly related pairs of parent and child nodes. That is, each variable (or node) is independent of all other variables except its own parental nodes given the information of the parent nodes. Let p(xi I xj. x, i) = p(xi 177,) be the (full) conditional distribution of x, given all the other variables. 

The joint probability distribution can then be expressed by the product of these full conditional distributions, applying the chain rule of conditional probabdities (Heckerman, 1995)  

These conditional independence relationships are used with three kinds of forms in the stmeture of the Bayesian network (Needham et al., 2008). 

---

Figure 7.1. A causal graph for risk analysis. The model depicted in this figure can be formalized using a Bayesian network (Ricci et al. 2006) A probabilistic framework interprets the model described in this figure as a Bayesian belief network or causal graph model. Each variable with inward-pointing arrows is interpreted as a random variable with a conditional probability distribution that depends only on the values of the variables that point into it. The essence of this approach to modeling and evaluating uncertain risks is to sample successively from the (often conditional) distribution of each variable, given the values of its predecessors. Algorithms exist to identify and validate possible causal structures.

It is difficult to cover the wide range of so many different network modeling approaches, but we will attempt to briefly introduce three widely used network modeling techniques in this chapter Boolean network (BN), Bayesian belief network, and metabolic network modeling methods. 

In the following, we will first deseribe the related regulatory issues. Then, we will present the basic approach to performing qualitative risk analysis based on Bayesian Belief Network (BBN) to identify unnecessary conservatism. A ease study dealing with the issue of independent verifieation and vahdation (IV V) will then be diseussed, followed by a eonclusion. 

A QA process model consists of elements representing software development staffs, software QA staffs, development activities, QA activities and documents generated. Our major concern, quality, is represented as defects density. Each element is represented as a node and is coimected based on its casual relation with other elements. Each node is further designated with 2 5 states representing its status, for example, the undesired event is represented as defect density at high status. In reality, the relation between QA process elements is not static and fixed, i.e., the influence of one node on the other node often exhibits probabilistic and interactive behavior. In order to represent the probabilistic behavior of QA process, we apply Bayesian Belief Network (BBN)(Jensen, 1996) technique. A typical QA process represented in BBN is shown in Fig 3. 

Characterization of uncertainties in the operation and economies of the proposed seawater desalination plant in the Gaza Strip was made by using a Bayesian belief network (BBN) approach [80]. In particular, the model was used to (1) characterize the different uncertainties involved in the RO process, (2) optimize the RO process reliability and cost, and (3) study how uncertainty in unit capital cost, unit operation and maintenance (O M) cost, and permeate quality was related to different input variables. The minimum specific capital cost was found to be 0.224 0.064 US /m, and the minimum O M cost was found to be 0.59 0.11 US /m. This unit cost was for a production capacity of 140,000 mVday. 

As we might expeet by this stage in the book, most of the usual classification/regression algorithms have been applied to the duration predietion problem. These include decision trees [372], neural networks for phone prediction [109], [157], genetic algorithms [319] and Bayesian belief networks [182], Comparative studies of deeision trees and neural networks found little difference in accuracy between either approaeh, [473], [187], [72],... 

Goubanova, O., and King, S. Predicting consonant duration with Bayesian belief networks. In Proceedings of the Interspeech 2005 (2005). 

Another type of dependency is that which results from some sort of causal mechanism. Such causality is often represented in Data Mining by using Bayesian Belief Networks which discover and describe. Such causal models allow us to predict consequences, even when circumstances change. If a rule just describes an association, then we cannot be sure how robust or generalizable it will be in the face of changing circumstances. 

A key feature of Bayesian Belief Networks is that they discover and describe causality rather than merely identifying associations as is the case in standard Statistics and Database Technology. Such causal relationships are represented by means of DAGs (Directed Acyclic Graphs) that are also used to describe conditional independence assumptions. Such conditional independence occurs when two variables are independent, conditional on another variable. 

Bouissou, M. Pourret, O. 2003. A Bayesian belief network based method for performance evaluation and troubleshooting of multistate systems. International Journal of Reliability, Quality and Safety Engineering, 10(4) 407- 16. 

Neil, M., Fenton, N., Forey, S. Harris, R, 2001. Using Bayesian belief networks to predict the reliability of military vehicles. Computing Control Engineering Journal, 12(1) 11-20. 

Evaluations of Safety Interventions Applications for Bayesian Belief Networks, Technical Report AHFD-05-13/NASA-05-04, prepared for NASA Langley Research Centre. 

Application of a d)mamic fault diagnostic method using Bayesian belief networks on a fuel rig system... 

ABSTRACT Bayesian Belief Networks are probabilistic models that are particularly suited to applications where new evidence can be introduced on the variables. By means of Bayes theorem, the probability associated to events can be updated following observations and new information available. This feature has advantages in system fault diagnostic processes where sensors on the system provide evidence for the state of a system. Where sensors indicate that the behavior of the monitored variable deviates from that expected, the information is used to find the possible causes of a fault. 

Bayesian Belief Networks or, simply, Bayesian Networks (BNs), represent a powerful modeUing tool for... 

Lampis M. Andrews J.D. 2008. Bayesian Belief Networks for System Fault Diagnostics. Quality and Reliability Engineering International (published online Nov 4, DOI 10.1002/qre.978). 

Lampis M. Andrews J.D. 2009. Use of Bayesian Belief Networks in a Dynamic Fault Diagnostic application, (to appear in the) Proceedings of the 18th ARTS (Advances in Reliability Technology Symposium) 21-23 April. 

Droguett, E. A. L., Moura, M. C., Jacinto, C. M. C., Silva Jr, M. E, Garcia, P. A. A. (2007). A semi-markov model with a Bayesian belief network based human reliability modeling for availability assessment of downhole optical monitoring systems. In Aven, Teije and Vinnem, J. E. (ed), Risk, Reliability and Societal Safety, Proc. European Safety and Reliability Conference, 25-27June 2007. Norway, Stavanger. 

---