Bayesian networks. A probabilistic model for chronic obstructive pulmonary disease diagnosis and phenotyping

TABLE OF CONTENTS

Thesis Abstract

ACKNOWLEDGEMENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS

CHAPTER 1: INTRODUCTION
1.1 General Introduction
1.2 Problem Statement
1.3 Objectives of the Study
1.3.1 Specific Objectives
1.4 Motivation
1.5 Research Questions
1.6 Definition of Terms
1.7 Research Scope
1.8 Thesis Organization

CHAPTER 2: LITERATURE REVIEW
2.1 Introduction
2.2 Theories and Background
2.1 Asthma
2.2 COPD
2.2.1 Chronic Bronchitis
2.2.2 Emphysema
2.2.3 COPD Risk Factors
2.2.4 The GOLD
2.2.5 Key Indicators for Considering a Diagnosis of COPD
2.2.6 Spirometry Reference Standard
2.2.7 Differential Diagnosis of COPD
2.2.8 COPD under Diagnosis and Misdiagnosis
2.3 Probabilistic Graphical Model (PGM)
2.3.1 Introduction to Probabilistic Reasoning
2.3.2 Basic Probability Calculus
2.3.3 Bayesian Networks
2.4 Reliability Statistics
2.5 Relevant Literature to this Topic
2.5.1 Case-Based Reasoning (CBR)
2.5.2 Fuzzy Expert System
2.5.3 Neural Networks
2.5.4 Support Vector Machine

CHAPTER 3: METHODOLOGY
3.1 Type of Study
3.2 The Study Population
3.3 Sample Size and Sampling Method
3.4 Research Environment
3.5 Data Collection Instrument
3.5.1 Checklist Validity
3.5.2 Checklist Reliability
3.6 Data Collection Method
3.7 Data Analysis
3.8 Research Variables
3.8.1 Creating the Bayesian network
3.8.2 Network Instantiation
3.9 Exact Inference in Bayesian Networks
3.9.1 Inference through Factor Elimination
3.9.2 Inference through Conditioning
3.9.3 Model Evaluation
3.9.4 F1Score
3.10 cardDiag Application
3.11 Limitations
3.12 Ethical Issues
3.13 Chapter Summary

CHAPTER 4: RESULTS
4.1 Introduction
4.1.1 Description of Our Data Set
4.2 Bayesian Network Accuracy
4.3 Results from the Test Data Set
4.3.1 Results Using the Bayesian Network
4.3.2 Results Using Levenberg-Marquardt (LM) Algorithm
4.3.3 Summary of the Classification Results for both Models
4.3.4 Evaluation of our Bayesian Network Model
4.3.5 cardDiag Application
4.4 Reliability Analysis Results
4.4.1 A Summary of COPD and Allergy Variable
4.4.2 Age Ranges for COPD Patients

CHAPTER 5: DISCUSSION, CONCLUSIONS, AND SUGGESTIONS
5.1 Discussion
5.1.1 Comparison of the Result from the BN and LM Algorithm
5.1.2 cardDiag Results
5.1.3 Essential Variables in COPD Diagnosis
5.2 Conclusion
5.3 Suggestions based on Research Results
5.3.1 Targeted Case Finding
5.3.2 Awareness Campaigns
5.4 Recommendations for Future Research
5.4.1 Large Equal Sample for each Phenotype
5.4.2 More Diagnosis and Screening Devices

6 REFERENCES

7 APPENDIX

8 BIODATA OF THE AUTHOR

9 LIST OF PUBLICATIONS

Thesis Abstract

A PROBABILISTIC MODEL FOR CHRONIC OBSTRUCTIVE PULMONARY DISEASE DIAGNOSIS AND PHENOTYPING USING BAYESIAN NETWORKS

By

Amos Otieno Olwendo

MARCH 2014

ABSTRACT

The World Health Organization (WHO) lists COPD as the fourth leading cause of the death worldwide yet the disease is preventable. Smoking of tobacco products, alpha-1-antitrypsin (AAt), and air pollution are the major risk factors associated with the development and progression of this disease. COPD is usually either misdiagnosed or under-diagnosed due to a number of factors including the slow progression of the development of its symptoms. Besides, differential diagnosis is usually applied during diagnosis because differentiating COPD patients from those with say chronic Asthma may not be an easy task. Previous researchers have used pulmonary function test results to diagnose COPD. However, this research provides a model for diagnosing and classifying COPD based on phenotypes; General COPD, Chronic bronchitis, Emphysema, and the Asthmatic COPD using a Bayesian network (BN). A BN is a probabilistic modelling tool composed of random variables and the relationships of such variables is based on probabilities that maximize certain outcomes. We validated our BN model using a neural network model based on the Levenberg- Marquardt (LM) algorithm. Results show that the BN model achieved an overall classification of 98.75% for our test cases. Furthermore, F1 score results also show that the BN is a better model for COPD classification in comparison to the LM algorithm.

ACKNOWLEDGEMENTS

This thesis owes a considerable debt of gratitude to a number of people who contributed to its completion and those who have influenced our thinking and behaviour over the years. I would like to thank all the faculty members whose guidance and support have made this study a success. I am specifically grateful to Dr. Hussein Arab-Alibeik for his support during the initial stages of this thesis including the completion of the thesis proposal. Moreover, I dearly thank Dr. Khosrow Agin, a respiratory system specialist for his immense knowledge, time, and generosity by which means he has contributed to the success of this research. Finally, I would like to thank Dr. Leila Shahmoradi for her guidance and support during the completion of this thesis. All your ideas and inspirations have made this study a success and a wonderful experience to me. Besides, I would like to thank my colleagues at the Medical Informatics department for their support to me as an International student here at Tehran University of Medical Sciences. Specifically, I would like to thank Parisa for translating our initial questionnaire to Farsi. Last but not least, I would like to offer my gratitude to the International Student family for the times we have had together as students here at TUMS. Thank you all.

LIST OF TABLES

Table

2.1: Indicators for considering a COPD diagnosis

3.1: Summary of the variables used in creating our BN model

3.2: The distribution of the reality and the predicted results of any data set

4.1: BN Classification results of 40 Asthma test cases

4.2: BN Classification results of 40 COPD test cases

4.3: Neural Network Classification results of 40 Asthma test cases

4.4: Neural Network Classification results of 40 COPD test cases

4.5: Summary of the Classification Results for both Models

4.6: A summary of accuracy, precision, recall and the F1 score

4.7 A summary of reliability analysis using Cronbach’s Alpha

4.8: A summary of Item-Total Statistics from reliability analysis

4.9: A Summary of COPD and Allergy Variable

4.10: Age Ranges for COPD Patients

5.1: Variables determined to be essential in this CDSS Design

LIST OF FIGURES

Figure

2.1: The GOLD standard for classifying COPD Severity

2.2: Considerations for performing a spirometry

2.3: An example diagram of a Naïve Bayes Classifier

2.4: A sketch diagram of variables in a Bayesian network

2.5: A sketch diagram showing the design of the Noisy-Or micro model

2.6: A pruned Bayesian network after bypassing non-essential variables

3.1: A summary of reliability analysis using Cronbach’s Alpha

3.2: A summary of the data collection procedures used in this research

3.3: Initial BN before applying various micro model design techniques

3.4: Initial BN designed using the Noisy-Or model design techniques

3.5: A Final Noisy-Or design for our BN model

3.6: Network of the variables and their corresponding parameters

3.7: A screen snapshot of the cardDiag Application

3.8: A group plot of the cardDiag Application

4.1: An MPE instantiation given evidence patient has no sputum

4.2: A MAP instantiation given the evidence that a patient has no sputum

4.3: Neural Network Application implemented using the LM algorithm

4.4: A group plot of the cardDiag Application

LIST OF ABBREVIATIONS

Abbildung in dieser Leseprobe nicht enthalten

CHAPTER 1

INTRODUCTION

1.1 General Introduction

The advancements in artificial intelligence (AI) and machine learning (ML) research have led to a greater percentage of human life being channelled towards the control and monitoring using automated processes [1]. Such advances are minimizing the need for human input in a number of applications. For example, image processing techniques have improved the spectrum of security cameras from the times when such cameras only recorded the happenings within their range of view to surveillance cameras that can actually determine a suspicious behavior.

On the other hand, probabilistic reasoning has a lot to offer in medical diagnosis given the nature of the many challenges encountered. The main task in probabilistic reasoning is to identify all relevant variables x 1 ,…,x n in a given environment and create a model p(x 1 ,…,x n ) that mimics the interactions of such variables [2]. Inference is then performed by introducing evidence and then calculating probabilities of interest based on the given set of evidence. As a matter of fact, medical diagnosis is prone to incomplete vague observations made by patients that are sometimes exaggerated yet the physicians are expected to achieve the best outcomes for the patient [1-2]. In addition, a physician’sdecisionsaboutapatient’s nexttestfollowsbasedonobserved current status of events.

Probabilistic reasoning does tend to offer an ability to capture and adequately represent patient symptoms, risk factors and other important variables to be used for disease diagnosis by assigning a degree of belief to every observation. However, one central thing to always remember is that computational techniques cannot be completely used to eliminate the human expertise in the medical domain. We still need the expertise.

1.2 Problem Statement

Chronic Obstructive Pulmonary Disease (COPD) is the fourth leading cause of death in both developing and developed nations yet the disease is preventable. WHO projections state that COPD mortalities are expected to increase by about 30% in the next 10 years [3-4]. Moreover, WHO reports that 90% of COPD deaths are realized in middle to low-income countries [5]. Furthermore, reports from developed countries show that COPD is prone to misdiagnosis and under- diagnosis yet healthcare facilities in such countries are well stocked with equipment necessary for an accurate diagnosis of COPD. Therefore, we believe the actual burden of COPD in middle and low-income countries are under estimated since such countries do not have adequate equipment and health services hence a number of COPD cases are never registered.

1.3 Objectives of the Study

This thesis entails the development of a Bayesian network model for COPD diagnosis that can as well classify COPD cases based on the phenotype. In addition, this research also attempts to understand the relationships between COPD phenotypes. Besides, we also determined whether or not a given patient case has Asthma given the fact that a successful diagnosis of COPD requires a differential diagnosis of Asthma. Finally, we developed cardDiag, a C++ application that determines the maximum likelihood estimation for each patient’s case. The results from cardDiag based on our test data set, graphically classify COPD phenotypes and Asthma in a manner that more or less is a pictorial relationship of these diseases as in a typical Venn diagram. The relationships of COPD phenotypes and Asthma show overlaps that exist amongst the phenotypes and could probably be due to the nature or magnitude of the air obstruction. However, we have not specifically done any work to determine the relationships of air obstruction and COPD phenotypes.

1.3.1 Specific Objectives

This research attempts to achieve the following objectives in the descending order:-

- To determine whether or not a patient is suffering as a result of COPD
- To determine the relative probabilities of each of the COPD phenotypes for each patient case
- To determine whether or not a patient has COPD and Asthma
- To determine the essential variables and parameters for use in developing a clinical decision support system (CDSS) for diagnosing COPD and Asthma

1.4 Motivation

This research is meant to use probabilistic modeling techniques to design a software application to be used in diagnosing and classifying COPD cases according to phenotypes. As a matter of fact, as previously discussed, COPD is prone to either under-diagnosis and or misdiagnosis. Therefore, we believe COPD’s burden worldwide is underestimated since WHO reports that 90% of COPD mortalities are experienced in middle and low-income countries where the quality of healthcare services is characteristically low. Moreover, after a careful review of literature on this topic, we did not identify a single research that used computational techniques to classify COPD cases based on phenotypes. However, we hope to classify COPD cases based on the corresponding phenotype.

1.5 Research Questions

This research is meant to provide answers to the following questions:

- Can probabilistic graphical modeling techniques be adequately employed in COPD diagnosis?
- Can we possibly attempt to determine the probabilities of classifying COPD patients as either suffering from one or a combination of COPD phenotypes?
- Can a probabilistic model determine whether a COPD patient is also suffering from Asthma?
- Can probabilistic modeling techniques adequately determine the essential variables and parameters to use for diagnosing COPD and Asthma?

1.6 Definition of Terms

- Model – is a declarative representation that mimics the functioning of the actual represented system.
- COPD phenotype – is the act of clustering COPD manifestations based on appearance, function, and behavior.
- GOLD – is an abbreviation for Global Initiative for Chronic Obstructive Lung Diseases. This is a global initiative responsible for the diagnosis, management, and prevention of COPD disease.
- BODE (Body mass index, Obstruction, Dyspnea, and Exercise) – is a tool used by healthcare professionals to predict COPD mortality.

1.7 Research Scope

We chose to use a Bayesian network because BNs are popularly known tools for modeling and making accurate complicated inferences thus a good tool for modeling intelligent CDSS [1]. For thepurposeocf ompletingaMaster’sthesis within the given time frame, this research is meant to design a software model for a CDSS for use in the diagnosis of COPD.

However, developing the model is not the end of this research as the author looks forward to implementing the next phase of this research using any of the Machine Learning techniques to deliver a final product that could be used in any clinic for the diagnosis of COPD. Such a task would require a number of resources including both time and money. Therefore, the author hopes to bring this expectation to an end as a fulfillment for the requirements of a PhD thesis project.

1.8 Thesis Organization

This thesis is organized into five chapters with chapter 2 covering the theoretical setting used in the design of this model; both the relevant theories on COPD and Bayesian networks. Chapter 3 covers the methodology of this study where we illustrate the step-by-step details of the content and the development of our model. Moreover, we used Chapter 4 to discuss the results we achieved from experimental analysis and test data set results from our model. Furthermore, we discuss the results of this model in comparison to the results achieved by the validating model. In addition, we also developed a C++ application, cardDiag; that enables us to graphical depict the relations of the COPD phenotypes. Finally, we give concluding remarks in Chapter 5 which includes the discussions of the results we have achieved.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

In this chapter, we discuss the theoretical background and the anatomies of COPD and Asthma, with focus on the details of their diagnosis. In addition, we discuss how the physicians identify COPD and Asthma in their daily medical practice. We introduce probabilistic graphical modelling as a tool for illustrating the relationship of events that lead to certain effects. Also, we give an abstract background on the model and how a Bayesian network fits as a tool for modelling and its potential for providing solutions to COPD diagnosis. Furthermore, we take our audience through the key features of a Bayesian network as a modelling apparatus from its representation to inference methods and techniques.

2.2 Theories and Background

Medical diagnosis is both a science and an art as a physician’sabilityto accurately diagnose a given disease from patient’s evaluation data and observationsmainlydependentonthe physician’sexperienceandprobably other specific skills. One of the challenges encountered during medical diagnosis is the situation where some diseases have a lot of resemblance and an inexperienced physician will most likely end up spending extra time and resources compared to their experienced counterparts. In such a case, a differential diagnosis of the relevant diseases is usually considered to ensure correct end results. For example, COPD and chronic Asthma share a lot of characteristics, therefore, diagnosing and classifying COPD may be an uphill task since a physician will not only have to focus on COPD itself but also consider differentiating COPD and Asthma cases.

2.1 Asthma

Asthma is a disorder characterized by chronic airway obstruction, hypersensitivity, and airway inflammation as a result of encounters to various stimuli [6-8]. The typical airway obstruction you can see in Asthma, results from a hypersensitivity-induced inflammation causing swelling and increased mucus production. Furthermore, Asthma and COPD are usually separated by examining the airflow obstruction after administering a bronchodilator which is known to be completely reversible. Asthma effects are experienced both during exhalation and inhalation [9].

Asthma patients experience symptoms at different degrees, from a minor nuisance to life-threatening attacks. Common Asthma symptoms include chest tightness and shortness of breath (dyspnea), wheezing, sputum production and coughing. In addition, a combination of symptoms like shortness of breath and chest tightness cause discomfort that result in trouble sleeping at night. Asthma may also be a risk factor for the development of COPD [7, 10]. Unfortunately, as of today, there is no cure for Asthma. Instead, treatments provide relief to the patients.

2.2 COPD

COPD is a disease that results in an inflammation and obstruction of the small airways in the lungs [6-7]. COPD disease may result in less airflow, due to one or more of the following reasons; the airways and alveoli losing their elastic quality, the walls between the alveoli being destroyed, the walls of the airways becoming thick and inflammated, or the airways secreting a lot of mucus leading to their blockage [6]. The major constituent diseases in COPD include chronic bronchitis, and Emphysema [6-7, 11]. However, a COPD patient may have a combination of the constituent diseases with one or so being dominant.

Based on a clinical perspective, identifying phenotypes of a disease means differentiating the disease cases based on appearance, function, and behavior. That is, COPD phenotypes classify patients into subgroups with distinct attributes [12]. A patient may be suffering as a result of a combination of COPD phenotypes or just any single one. Therefore, a patient may be sick due to chronic bronchitis and emphysema which are the main constituent diseases in COPD. Moreover, in any case a patient has a combination of chronic bronchitis and emphysema with both dominant, such a combination is classified as General COPD. Furthermore, a mix of any of the other COPD phenotypes with asthma is referred to as the asthmatic COPD phenotype [6, 13-15].

2.2.1 Chronic Bronchitis

Chronic bronchitis is one of the constituent diseases in COPD and it is characterized by a constant irritation and or inflammation of the linings of the bronchial tube [6-7, 11]. Chronic bronchitis develops from its acute form and is usually caused by smoking and exposure to irritations.

Moreover, chronic bronchitis symptoms include chronic cough and sputum (mucus) production that starts out as colorless and the coloration of the sputum turns yellowish and even bloody as the disease progresses. As the bronchial epitheliumgetsmoreinflameda,person’schancesof getting other infections increase. The continual inflammation overtime leads to the damage to the walls of the airways. This further leads to narrowing of the airways as they get filled with mucus.

2.2.2 Emphysema

On the other hand, Emphysema as one of the constituent diseases in COPD is characterized by the destruction of the air sacs (alveoli) leading to shortness of breath (dyspnea) [11]. The alveolus is comprised of minute veins and capillaries through which gaseous exchange take place during the intake of O2 and the exhalation of CO2. Emphysema mainly develops from a background of heavy smoking of cigarette, marijuana, or a substantial exposure to irritants. Even so, as the disease progresses; the patient tends to avoid activities that lead to shortness of breath.

2.2.3 COPD Risk Factors

The development and progression of COPD is as a result of a person getting exposed to any or a combinations of the risk factors [6-7]. Research has shown thaatperson’schancesodf evelopingairwaysobstructioni ncreases with age. Actually, COPD is sometimes known as an old-age disease since it is only common amongst persons advanced in age.

Experience also shows that young heavy smokers also develop COPD [4, 16- 17]. Cigarette smoking is one of the well known causes of COPD. Also, there is evidence showing that non-smokers may also develop chronic airflow inflammation. Moreover, amongst people with the same smoking history, not all will develop COPD at the same rate maybe as a result of the differences in the genetic factors, age, or gender [7]. However, COPD cases are projected to rise over the next 10 years due to an increasing smoking habit especially among women and teenagers [4, 10].

Moreover, environmental pollution as a result of industrial wastes and auto engine emissions into the atmosphere and other forms of air pollution; indoor air pollution from second-hand smoke, burning of wood, animal dung also pose a risk for developing COPD [6-7]. Besides, occupational exposures to organic and inorganic dusts, chemical agents and fumes may as well contribute to the development of COPD. Exposure to such agents may lead to or may be accelerated by allergic reactions that result to the inflammation of the airways.

Nevertheless, all the aforementioned risk factors do not affect different individuals exposed at the same time to the same magnitude. Therefore, the development and progression of COPD is experienced differently by different individuals and genetic factors seem to play a greater role in COPD disease development and progression [6-7].

People with a genetic predisposition that is said to lead to the development of COPD have a deficiency of alpha-1 antitrypsin (AAt), a protein made in the liver [11]. However, as of today, there exists a therapy mechanism for encountering this deficiency [18]. Thus, people susceptible of developing COPD as a result of the AAt deficiency should be counted to have a solution to such a problem. However, we can easily observe that there are a number of factors that could lead to the development of COPD some of which are not probably yet identified or well reasoned out.

2.2.4 The GOLD

The Global Initiative for Chronic Obstructive Lung Disease (GOLD); is a global strategy for the diagnosis, management and prevention of COPD [7-8]. For that reason, GOLD has come up with the following standard indicators to be used for considering a COPD diagnosis. These include; dyspnea (shortness of breath) that worsens over time, cough that becomes chronic over time, a pattern of sputum production (whether colored or colorless), a history of exposure to risk factors e.g. all forms of tobacco smoke, smoke from home cooking and heating fuels, and occupational dusts and chemicals, a family history of COPD. Additionally, the GOLDrecommendslookingintothepatient’smedicahl istoryof Asthma, allergy, respiratory system infections, co morbidities such as heart and lung diseases, being overweight or underweight, osteoporosis, musculoskeletal disorders etc. The GOLD standard has also established parameters for diagnosing and staging COPD from Spirometry test results using FEV1 and FVC [7-8]. Figure 2.1 below shows the stages of COPD based on the GOLD standards. All the literature identified to be related to this research topic used such classifications as shown in figure 2.1 below. However, this research is the first attempt to classify COPD cases based on the phenotype that a given patient is suffering from.

Figure 2.1: The GOLD standard for classifying COPD [7-8].

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2.2.5 Key Indicators for Considering a Diagnosis of COPD

The table below shows a standard reference standard of the key indicators for considering a diagnosis of COPD. These parameters are measured during patient evaluation which includes history taking of the social and past medical history, history of symptoms, vital signs etc. Usually, a patient with any of these indicators would then be referred to take a spirometry test. A spirometry test is the current standard used to establish a diagnosis of COPD [8, 10].

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Table 2.1: Indicators for considering a COPD diagnosis [8, 10]

2.2.6 Spirometry Reference Standard

Patients determined to be worthy of consideration for COPD diagnosis should be subjected to performing a spirometry test.

Figure 2.2: Considerations for performing a spirometry [8]. From : GOLD_Report_2011_Feb21. Accessed 10/09/2013.

2.2.7 Differential Diagnosis of COPD

COPD is usually accompanied by one or so co morbidities that are related to either smoking, respiratory infections, and or aging [6]. In the case of a disease having similar symptoms like in COPD, diagnosing COPD may become an uphill task. For example, a number of patients with chronic asthma are commonly misdiagnosed as COPD [19-21]. As a result, diagnosing COPD from related co morbidities needs the use of a systematic method that can adequately minimize the probabilities of COPD to negligible levels.

COPD is usually characterized by an onset in patients in their mid thirties or older and the symptoms and disease progression may depend on factors related to genetics, nutrition, and social behaviors such as smoking in addition to other causes [7-8, 10]. On the other hand, Asthma is well known to have genetic causes and a number of Asthma cases are known to start as early as during childhood with varying symptoms that worsen at night and or in the morning [6,9]. Therefore, Asthma patients tend to have a family history of the disease in one of their immediate family members in addition to experiencing allergic reactions like allergies to certain foods or substances like perfumes etc.

2.2.8 COPD under Diagnosis and Misdiagnosis

The progression of COPD is usually slow hence the symptoms become apparent in the advanced stages of the disease [22]. A number of persons suffering from moderate to severe COPD do not know that they are suffering from COPD as such persons tend to compensate for their breathing problems in addition to assuming that their health conditions are part of the natural aging process. As a result, COPD cases are prone to under-diagnosis and or misdiagnosis [19-21, 23]. This is as a result of a number of factors in addition to thepatient’slackofawarenessof COPDandotherchronicrespiratory complications.

2.3 Probabilistic Graphical Model (PGM)

A graphical model is a probabilistic model for which a graph denotes the restrictive dependence structure linking its random variables. Therefore, graphical modelling techniques are appropriate for creating models for disease diagnosis [1] because of their ability to handle vagueness which comes as a result of:

- Biased and or incomplete understanding of the situation or event or the world in question
- Noisy observations
- Phenomena not specifically included in the model in questions, and
- The randomness of events in daily life.

Moreover, probabilistic graphical modelling is strengthened by probability theory which offers a declarative representation to the vague events that offer great reasoning capabilities. Therefore, an amalgamation of graphical models and probability theory is more intuitive hence serves as a great tool for probabilistic reasoning.

There are two types of graphical models; the Bayesian Network (BN) and Markov Network (MN). However, this study is only focused on modelling and reasoning with Bayesian networks for COPD diagnosis given the fact that a Bayesian network gives a clear representation of the cause and effects as experienced in the case of diseases, risk factors and symptoms. As a result of their graphical representation, Bayesian networks are known as directed acyclic graphs (DAGs) given the fact that the connections from one variable to another is only in one direction and you cannot traverse through the network variables and end up where you started [1].

2.3.1 Introduction to Probabilistic Reasoning

In daily life, human decision making is usually accompanied by the use of known facts and or underlying assumptions. Moreover, the use of facts and assumptions in reasoning is based on our degree of belief in such an event either occurring or not. It is therefore the degree of belief we have in each event that makes it qualify as either a fact or an assumption [24]. A degree of belief can either be represented using the concept of fuzzy logic or probabilities. In this thesis, we will only consider representing degrees of belief as probabilities and manipulate degrees of belief using the laws of probability [1].

Probabilistically, a degree of belief in an event is adjustable either in an increasing or in a decreasing order depending on the level of assurance we have on the new evidence. On the other hand, assumptions are addressed as either believable or unbelievable depending on any new information we may have. As a result, assigning degrees of belief at different events is a better undertaking than relying on assumptions which tend to be in the true false format. Therefore, the levels of granularity in the degrees of belief have a means of controlling and guiding us against falling into some of the pitfalls that you could easily experience when working with assumptions.

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