Hybrid Bayesian networks (HBN) contain complex conditional probability distributions (CPD) specified as partitioned expressions over discrete and continuous variables. The size of these CPDs grows exponentially with the number of parent nodes, and when using discrete inference methods, it results in significant execution time and space inefficiency. To reduce the CPD size, a binary factorization (BF) algorithm can be used to decompose the statistical or arithmetic functions in the CPD by factorizing the number of connected parent nodes into sets of size two. However, the BF algorithm was not designed to handle partitioned expressions. Therefore, we propose a new stacking factorization (SF) algorithm to decompose partitioned expressions. The SF algorithm creates intermediate nodes to incrementally reconstruct the conditional densities in the original partitioned expression, ensuring that no more than two continuous parent nodes are connected to each child node in the resulting HBN. It generally applies to both discrete and continuous child nodes with complex partitioned expressions. When we combine SF with a dynamic discretization (DD) inference algorithm, we achieve a significant improvement in inference efficiency. Experimental results demonstrate that the combination of SF and DD can effectively manage HBNs with complex CPDs that may challenge other algorithms, which also outperform competing inference algorithms in accuracy.

The partition function Ƶ ${\boldsymbol{Ƶ}}$ is a normalization constant for normalizing all the distributions in probabilistic inference. Ƶ ${\boldsymbol{Ƶ}}$ is closely related to the log probability of evidence ( log p ( e ) $\mathrm{log}\unicode{x0200A}p(e)$ ) for Bayesian networks (BNs), which plays an important role in many applications, such as parameter learning, classification, and clustering. However, evaluating Ƶ ${\boldsymbol{Ƶ}}$ and log p ( e ) $\mathrm{log}\unicode{x0200A}p(e)$ for BNs containing both discrete and continuous variables (known as hybrid Bayesian networks [HBNs]) is generally difficult for analytical and simulation-based solutions. This paper describes the work addressing the problem of estimating the partition function for HBNs when exact methods become inefficient or numerically unstable. We use dynamic discretization to provide a discretized version of the model and then perform region-based free energy optimization to obtain the log p ( e ) $\mathrm{log}\unicode{x0200A}p(e)$ for the HBN efficiently, which is called the DDJT-region partition function (RPR) algorithm. It is novel since the region-based methods have previously only been applied to discrete BNs. We show that the log p ( e ) $\mathrm{log}\unicode{x0200A}p(e)$ obtained by DDJT-RPR is exact for discretized models and with space complexity equal to that in marginal inference, thus generally applicable to a wide range of applications. To demonstrate these properties, we combine the DDJT-RPR algorithm with an improved expectation-maximization algorithm to learn Gaussian mixture models (GMMs), for applications requiring data only and with prior knowledge, which performed more robustly than conventional GMM learning algorithms.

Bayesian networks (BNs) have received increasing research attention that is not matched by adoption in practice and yet have potential to significantly benefit healthcare. Hitherto, research works have not investigated the types of medical conditions being modelled with BNs, nor whether there are any differences in how and why they are applied to different conditions. This research seeks to identify and quantify the range of medical conditions for which healthcare-related BN models have been proposed, and the differences in approach between the most common medical conditions to which they have been applied. We found that almost two-thirds of all healthcare BNs are focused on four conditions: cardiac, cancer, psychological and lung disorders. We believe there is a lack of understanding regarding how BNs work and what they are capable of, and that it is only with greater understanding and promotion that we may ever realise the full potential of BNs to effect positive change in daily healthcare practice.

Performing efficient inference on high dimensional discrete Bayesian Networks (BNs) is challenging. When using exact inference methods the space complexity can grow exponentially with the tree-width, thus making computation intractable. This paper presents a general purpose approximate inference algorithm, based on a new region belief approximation method, called Triplet Region Construction (TRC). TRC reduces the cluster space complexity for factorized models from worst-case exponential to polynomial by performing graph factorization and producing clusters of limited size. Unlike previous generations of region-based algorithms, TRC is guaranteed to converge and effectively addresses the region choice problem that bedevils other region-based algorithms used for BN inference. Our experiments demonstrate that it also achieves significantly more accurate results than competing algorithms.

The “leaky noisy-OR” function is a common and popular method used to simplify the elicitation of complex conditional probability tables in Bayesian networks involving Boolean variables. It has proven to be useful for approximating the required relationship in many real-world situations where there is a set of two or more variables that are potential causes of a single effect variable. However, one of the properties of leaky noisy-OR is Conditional Inter-causal Independence (CII). This property means that the `explaining away' behavior-one of the most powerful benefits of BN inference-is not present when the effect variable is observed as false. Yet, for many real-world problems where the leaky noisy-OR has been considered, this behavior would be expected, meaning that leaky noisy-OR is deficient as an approximation of the required relationship in such cases. There have been previous attempts to adapt the noisy-OR to resolve this problem. However, they require too many additional parameters to be elicited. We describe a simple but powerful extension to leaky noisy-OR that requires only a single additional parameter. While it does not solve the CII problem in all cases, it does resolve most of the explaining away deficiencies that occur in practice. The problem and solution is illustrated using an example from intelligence analysis.

Numerous methods have been proposed to simplify the problem of eliciting complex conditional probability tables in Bayesian networks. One of the most popular methods -“Noisy-OR”- approximates the required relationship in many real-world situations between a set of variables that are potential causes of an effect variable. However, the Noisy-OR function has the conditional inter-causal independence (CII) property which means that `explaining away' behavior-one of the most powerful benefits of BN inference-is not present when the effect variable is observed as false. Hence, for many real-world problems where the Noisy-OR has been used or proposed, it may be deficient as an approximation of the required relationship. However, there is a very simple alternative solution, namely to define the variables as ranked nodes and to use the ranked node weighted average function. This does not have the CII property-instead, we prove it has the conditional anti-correlation property required to ensure that explaining away works in all cases. Moreover, ranked node variables are not restricted to binary states, and hence provide a more comprehensive and general solution to Noisy-OR in all cases.

One of the greatest impediments to the use of probabilistic reasoning in legal arguments is the difficulty in agreeing on an appropriate prior probability for the ultimate hypothesis, (in criminal cases this is normally "Defendant is guilty of the crime for which he/she is accused"). Even strong supporters of a Bayesian approach prefer to ignore priors and focus instead on considering only the likelihood ratio (LR) of the evidence. But the LR still requires the decision maker (be it a judge or juror during trial, or anybody helping to determine beforehand whether a case should proceed to trial) to consider their own prior; without it the LR has limited value. We show that, in a large class of cases, it is possible to arrive at a realistic prior that is also as consistent as possible with the legal notion of 'innocent until proven guilty'. The approach can be considered as a formalisation of the 'island problem' whereby if it is known the crime took place on an island when n people were present, then each of the people on the island has an equal prior probability 1/n of having carried out the crime. Our prior is based on simple location and time parameters that determine both a) the crime scene/time (within which it is certain the crime took place) and b) the extended crime scene/time which is the 'smallest' within which it is certain the suspect was known to have been 'closest' in location/time to the crime scene. The method applies to cases where we assume a crime has taken place and that it was committed by one person against one other person (e.g. murder, assault, robbery). The paper considers both the practical and legal implications of the approach. We demonstrate how the opportunity prior probability is naturally incorporated into a generic Bayesian network model that allows us to integrate other evidence about the case.

We show that existing Bayesian network (BN) modelling techniques cannot capture the correct intuitive reasoning in the important case when a set of mutually exclusive events need to be modelled as separate nodes instead of states of a single node. A previously proposed ‘solution’, which introduces a simple constraint node that enforces mutual exclusivity, fails to preserve the prior probabilities of the events, while other proposed solutions involve major changes to the original model. We provide a novel and simple solution to this problem that works in all cases where the mutually exclusive nodes have no common ancestors. Our solution uses a special type of constraint and auxiliary node together with formulas for assigning their necessary conditional probability table values. The solution enforces mutual exclusivity between events and preserves their prior probabilities while leaving all original BN nodes unchanged.

An important challenge in the field of law is the attribution of responsibility and blame to individuals and organisations for a given harm. Attributing legal responsibility often involves (but is not limited to) assessing to what extent certain parties have caused harm, or could have prevented harm from occurring. This paper presents a causal framework for performing such assessments that is particularly suitable for the analysis of complex legal cases, where the actions of many parties have had a direct or indirect effect on the harm that did occur. This framework is evaluated by means of a case study that applies it to the Baby P. case, a high-profile case of child abuse leading to the death of a child that has been the subject of a number of public inquiries in the UK. The paper concludes with a discussion of the framework, including a roadmap of future work and barriers to adoption.

We present a Bayesian network (BN) model for forecasting Association Football match outcomes. Both objective and subjective information are considered for prediction, and we demonstrate how probabilities transform at each level of model component, whereby predictive distributions follow hierarchical levels of Bayesian inference. The model was used to generate forecasts for each match of the 2011/2012 English Premier League (EPL) season, and forecasts were published online prior to the start of each match. Profitability, risk and uncertainty are evaluated by considering various unit-based betting procedures against published market odds. Compared to a previously published successful BN model, the model presented in this paper is less complex and is able to generate even more profitable returns.

Reducing the computational complexity of inference in Bayesian Networks (BNs) is a key challenge. Current algorithms for inference convert a BN to a junction tree structure made up of clusters of the BN nodes and the resulting complexity is time exponential in the size of a cluster. The need to reduce the complexity is especially acute where the BN contains continuous nodes. We propose a new method for optimizing the calculation of Conditional Probability Tables (CPTs) involving continuous nodes, approximated in Hybrid Bayesian Networks (HBNs), using an approximation algorithm called dynamic discretization. We present an optimized solution to this problem involving binary factorization of the arithmetical expressions declared to generate the CPTs for continuous nodes for deterministic functions and statistical distributions. The proposed algorithm is implemented and tested in a commercial Hybrid Bayesian Network software package and the results of the empirical evaluation show significant performance improvement over unfactorized models.

Bayesian networks, which can combine sparse data, prior assumptions and expert judgment into a single causal model, have already been used to build software effort prediction models. We present such a model of an extreme programming environment and show how it can learn from project data in order to make quantitative effort predictions and risk assessments without requiring any additional metrics collection program. The model's predictions are validated against a real world industrial project, with which they are in good agreement.

Although Bayesian Nets (BNs) are increasingly being used to solve real world risk problems, their use is still constrained by the difficulty of constructing the node probability tables (NPTs). A key challenge is to construct relevant NPTs using the minimal amount of expert elicitation, recognising that it is rarely cost-effective to elicit complete sets of probability values. We describe a simple approach to defining NPTs for a large class of commonly occurring nodes (called ranked nodes). The approach is based on the doubly truncated Normal distribution with a central tendency that is invariably a type of weighted function of the parent nodes. In extensive real-world case studies we have found that this approach is sufficient for generating the NPTs of a very large class of nodes. We describe one such case study for validation purposes. The approach has been fully automated in a commercial tool, called AgenaRisk, and is thus accessible to all types of domain experts. We believe this work represents a useful contribution to BN research and technology since its application makes the difference between being able to build realistic BN models and not.

A hybrid Bayesian network (BN) is one that incorporates both discrete and continuous nodes. In our extensive applications of BNs for system dependability assessment the models are invariably hybrid and the need for efficient and accurate computation is paramount. We apply a new iterative algorithm that efficiently combines dynamic discretisation with robust propagation algorithms on junction tree structures to perform inference in hybrid BNs. We illustrate its use on two example dependability problems: reliability estimation and diagnosis of a faulty sensor in a temporal system. Dynamic discretisation can be used as an alternative to analytical or Monte Carlo methods with high precision and can be applied to a wide range of dependability problems.

Recent work in applying causal modeling (Bayesian networks) to software engineering has resulted in improved decision support systems for software project managers. Once the causal models are built there are commercial tools that can run them. However, data to populate the models is typically entered manually and this is an impediment to their more widespread use. Hence, here we present a prototype tool for automatically extracting a range of relevant software metrics from popular project management and CASE tools. This information is used to populate Bayesian networks with the aim of providing better real world predictions of the risks associated with software costs, timescales and reliability.

Software metrics should support managerial decision making in software projects. We explain how traditional metrics approaches, such as regression-based models for cost estimation fall short of this goal. Instead, we describe a causal model (using a Bayesian network) which incorporates empirical data, but allows it to be interpreted and supplemented using expert judgement. We show how this causal model is used in a practical decision-support tool, allowing a project manager to trade-off the resources used against the outputs (delivered functionality, quality achieved) in a software project. The model and toolset have evolved in a number of collaborative projects and hence capture significant commercial input. Extensive validation trials are taking place among partners on the EC funded project MODIST (this includes Philips, Israel Aircraft Industries and QinetiQ) and the feedback so far has been very good. The estimates are sensible and the causal modelling approach enables decision-makers to reason in a way that is not possible with other project management and resource estimation tools. To ensure wide dissemination and validation a version of the toolset with the full underlying model is being made available for free to researchers.

The authors describe a number of results from a quantitative study of faults and failures in two releases of a major commercial software system. They tested a range of basic software engineering hypotheses relating to: the Pareto principle of distribution of faults and failures; the use of early fault data to predict later fault and failure data; metrics for fault prediction; and benchmarking fault data. For example, we found strong evidence that a small number of modules contain most of the faults discovered in prerelease testing and that a very small number of modules contain most of the faults discovered in operation. We found no evidence to support previous claims relating module size to fault density nor did we find evidence that popular complexity metrics are good predictors of either fault-prone or failure-prone modules. We confirmed that the number of faults discovered in prerelease testing is an order of magnitude greater than the number discovered in 12 months of operational use. The most important result was strong evidence of a counter-intuitive relationship between pre- and postrelease faults; those modules which are the most fault-prone prerelease are among the least fault-prone postrelease, while conversely, the modules which are most fault-prone postrelease are among the least fault-prone prerelease. This observation has serious ramifications for the commonly used fault density measure. Our results provide data-points in building up an empirical picture of the software development process.

Many organizations want to predict the number of defects (faults) in software systems, before they are deployed, to gauge the likely delivered quality and maintenance effort. To help in this numerous software metrics and statistical models have been developed, with a correspondingly large literature. We provide a critical review of this literature and the state-of-the-art. Most of the wide range of prediction models use size and complexity metrics to predict defects. Others are based on testing data, the "quality" of the development process, or take a multivariate approach. The authors of the models have often made heroic contributions to a subject otherwise bereft of empirical studies. However, there are a number of serious theoretical and practical problems in many studies. The models are weak because of their inability to cope with the, as yet, unknown relationship between defects and failures. There are fundamental statistical and data quality problems that undermine model validity. More significantly many prediction models tend to model only part of the underlying problem and seriously misspecify it. To illustrate these points the Goldilock's Conjecture, that there is an optimum module size, is used to show the considerable problems inherent in current defect prediction approaches. Careful and considered analysis of past and new results shows that the conjecture lacks support and that some models are misleading. We recommend holistic models for software defect prediction, using Bayesian belief networks, as alternative approaches to the single-issue models used at present. We also argue for research into a theory of "software decomposition" in order to test hypotheses about defect introduction and help construct a better science of software engineering.

There are many standards which are relevant for building safety- or mission-critical software systems. An effective standard is one that should help developers, assessors and users of such systems. For developers, the standard should help them build the system cost-effectively, and it should be clear what is required in order to conform to the standard. For assessors, it should be possible to objectively determine compliance to the standard. Users, and society at large, should have some assurance that a system developed to the standard has quantified risks and benefits. Unfortunately, the existing standards do not adequately fulfil any of these varied requirements. We explain why standards are the way they are, and then provide a strategy for improving them. Our approach is to evaluate standards on a number of key criteria that enable us to interpret the standard, identify its scope and check the ease with which it can be applied and checked. We also need to demonstrate that the use of a standard is likely either to deliver reliable and safe systems at an acceptable cost or to help predict reliability and safety accurately. Throughout the paper, we examine, by way of example, a specific standard for safety-critical systems (namely IEC 1508) and show how it can be improved by applying our strategy.

are glad that members of the software engineering communityhave taken-up our suggestion that metric validation needs publicdebate. With respect to specific issues raised by Basili et al. in theirletter, clearly, we disagree with their view that we are seriouslyimpeding the appropriate definition of attributes; we are glad tohave the opportunity to clarify some of our arguments.

In this paper we propose a framework for validating software measurement. We start by defining a measurement structure model that identifies the elementary component of measures and the measurement process, and then consider five other models involved in measurement: unit definition models, instrumentation models, attribute relationship models, measurement protocols and entity population models. We consider a number of measures from the viewpoint of our measurement validation framework and identify a number of shortcomings; in particular we identify a number of problems with the construction of function points. We also compare our view of measurement validation with ideas presented by other researchers and identify a number of areas of disagreement. Finally, we suggest several rules that practitioners and researchers can use to avoid measurement problems, including the use of measurement vectors rather than artificially contrived scalars.

Software measurement, like measurement in any other discipline, must adhere to the science of measurement if it is to gain widespread acceptance and validity. The observation of some very simple, but fundamental, principles of measurement can have an extremely beneficial effect on the subject. Measurement theory is used to highlight both weaknesses and strengths of software metrics work, including work on metrics validation. We identify a problem with the well-known Weyuker properties (E.J. Weyuker, 1988), but also show that a criticism of these properties by J.C. Cherniavsky and C.H. Smith (1991) is invalid. We show that the search for general software complexity measures is doomed to failure. However, the theory does help us to define and validate measures of specific complexity attributes. Above all, we are able to view software measurement in a very wide perspective, rationalising and relating its many diverse activities.>

Ideally, a measure of the security of a system should capture quantitatively the intuitive notion of ‘the ability of the system to resist attack’. That is, it should be operational, reflecting the degree to which the system can be expected to remain

Software metrication is a major problem in the engineering of software and software-related systems. To give coherence to attempts at metrication and to allow convincing statistical validation of these attempts it is proposed to axiomatise assumptions made about software parameters and ‘complexity’. The axioms are intrinsically related to the ‘decomposition hierarchy’ of programs, and this important notion is formalised in this paper for arbitrary programs.