What is a Bayesian Network?

A Bayesian belief network or Bayesian network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. If there is an arc from node A to another node B, then we say that A is a parent of B. If a node has a known value, it is said to be an evidence node. A node can represent any kind of variable, be it an observed measurement, a parameter, a latent variable, or a hypothesis. Nodes are not restricted to representing random variables; this is what is "Bayesian" about a Bayesian network.

A Bayesian network is a representation of the joint distribution over all the variables represented by nodes in the graph. Let the variables be X(1), ..., X(n). Let parents(A) be the parents of the node A. Then the joint distribution for X(1) through X(n) is represented as the product of the probability distributions p(X(i) | parents(X(i))) for i from 1 to n. If X has no parents, its probability distribution is said to be unconditional, otherwise it is conditional. Figure 1 is a simple BBN example.

Figure 1 BBN example

The usage of BBN can be divided into following four steps:

• Determine variable collection and variable domain
  This step is related to the problem domain. For the software reliability early prediction modeling, several complexity metrics are selected to build BBN.
• According to prior knowledge, determine network topology structure and probability distribution
  The total number of topology structure can be n! with n node. So the prior knowledge from domain expert is very important to get most possible topology. And if domain expert can also determine the reasonable probability distribution, then a lot of effort for collecting training data can also be reduced.
• Adjust topology structure and probability distribution using training data
  It's useful to introduce the prior knowledge from domain expert, but with the number of nodes increasing, such prior knowledge can be inaccurate. So, training data need to be collected to train the topology and probability distribution. Such adjustment is done by BBN learning algorithm. The purpose of learning is to use training data D and prior knowledge * to find the structure S whose posterior probability p(S|D, x) is maximal.
• Predict using BBN
  Here, prediction is reasoning. Bayesian reasoning is to calculate the conditional probability of nodes according to the information of other nodes. According to the dependency between the nodes, BBN reasoning can be divided into three types:
  • Causal reasoning, which is used to predict result according to the cause. It's a top-down reasoning.
  • Diagnosis reasoning, which is used to analyze cause according to result. It's a bottom-up reasoning.
  • Support reasoning, which is used to analyze the effect between each cause.

Comparing to the traditional regressive modeling, BBN has following advantages:

• BBN is highly related to Bayesian statistic. It's useful for establishing the relationship between knowledge and data.
• BBN can deal with the data set, which may be not self-contained or there are some noise inside, while traditional model can't.
• Because BBN can depict both the causality and probability, it's easy to be used with the process of decision-making.
• BBN uses graphic style to depict the dependency between data, so it's easy for understanding and explanation.

--which was extracted and refined from web material: Using Bayesian Networks for Water Quality Prediction in Sydney Harbour.

What have Bayesian Networks been used for?

Defect Detection - software debugging, safety and risk evaluation of complex systems

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Related topic: BNN (Bayesian Neural Network)

Detailed information can be found here.

Dealing with the Expert Inconsistency in Probability Elicitation ,

Knowledge Engineering for Bayesian Networks ,

Designing a Procedure for the Acquisition of Probability Constraints for Bayesian Networks ,

Generating Conditional Probabilities for Bayesian Networks: Easing the Knowledge Acquisition Problem ,

Induction of Bayesian Networks with a priori Domain Knowledge ,

Knowledge Engineering for Probabilistic Models: A tutorial ,

Using Sensitivity Analysis for Selective Parameter Update in Bayesian Network Learning ,

Gary D. Boetticher, Machine Learners Answer the 300-Billion-Dollar Question , University of Houston-Clear Lake

Fenton, N., and M. Neil, A Critique of Software Defect Prediction Research , IEEE Transaction on Software Engineering, Vol. 25, No. 5, 1999

S. Bibi, I. Stamelos, Software Process Modeling with Bayesian Belief Networks , IEEE Transaction on Software Engineering, Vol. 25, No. 5, 1999

S. Bibi, I. Stamelos, L. Angelis, Bayesian Belief Networks as a Software Productivity Estimation Tool , Department of Informatics, Aristotle University

Trevor Cockram, Gaining confidence in Software Inspection using a Bayesian Belief Model , Rolls-Royce plc and The Open University

Jilles van Gurp & Jan Bosch, Using Bayesian Belief Networks in Assessing Software Architectures , University of Karlskrona Ronneby, Department of Software Engineering and Computer Science

Hadar Ziv, Debra J. Richardson, Bayesian-network Confirmation of Software Testing Uncertainties , Department of Information and Computer Science, University of California, Irvine

Software cost estimation
Software cost estimation is the process of predicting the amount of effort required to build a software system. Models provide one or more mathematical algorithms that compute cost as a function of a number of variables. Size is a primary cost factor in most models and can be measuring using lines of code or function points. Models used to estimate cost can be categorized as either cost models or constraint models. COCOMO is an example of a cost model and SLIM is an example of a constraint model. Although criteria for evaluating a model have been suggested, there are some fundamental problems with existing models. Many models are available as automated tools.

Bayesian Elicitation of Experts Probabilities

The Probability Elicitation Tool

The Third Bayesian Modeling Applications Workshop During UAI-05, Uncertainty in Artificial Intelligence 2005 Edinburgh, Scotland, UK

the big guy who are doing some project related to BBN