Research Objectives and Framework

Based on the limitations mentioned above, this dissertation is motivated to investigate and develop the reasoning methods for Bayesian networks and influence diagrams with improved features.

The objectives of this dissertation are as follow.

1. Develop the reasoning models that can contain various kinds of Bayesian networks that may include crisp discrete nodes, continuous nodes, crisp parameters, fuzzy parameters, and decision nodes.

2. Introduce extra knowledge or constraints into the reasoning models, which can perform the propagation more efficiently and effectively.

3. Design the model that can complete diagnosis and suggest optimal treatment simultaneously, which can facilitate the performance in a business or medical decision support systems.

For the common base of research, this dissertation first defines a general Bayesian networks as follow.

Definition 1 General Bayesian networks.

A general Bayesian network (GBN) is a directed acyclic graph (DAG) representing the joint probability distribution of several sets of variables, including DN, CN, XN, L, P; that is . GBN= (DN, CN, XN, L, P), where

DN denotes a set of discrete random nodes;

CN denotes a set of continuous random nodes;

P denotes a set of parameters (probabilities);

XN denotes the decision node set;

L denotes a set of directed links between the nodes, such that L=(DN,CN,XN) × (DN,CN,XN) □

Based on the definition of GBN, we can induce several specific types of Bayesian networks. Consider a Bayesian network widely referred in Figure 1. Figure 1 represents the variables and their relationships from a medical problem. There are five random nodes, A, B, C, D, E. If all the random nodes in Figure 1 are discrete variables, and their probability distributions are crisp as in Table 1, then we can define a typical Bayesian network most common in the literatures, namely, BN1 = (DN, L, P).

If the parameters of the probability distributions are not crisp but fuzzy, for example, P(+b|+a) = ~x₁, P(+b|-a) = ~x₂, P(+c|+a) = ~x , P(+c|-a) = ₃ ~x₄, P(+d|+b,+c) = ~x , P(+d|-b, ₅ +c) = ~x , P(+d|+b, -c) = ₆ ~x , and P(+d|-b, -c) = ₇ ~x , then we can define the second type of ₈ Bayesian networks, BN2 in the form of BN2= (DN, L,P~

), where the parameter set turns into fuzzy.

E B

A

C

D

Metastatic cancer Increased total

serum calcium Brain tumor

Severe headaches Coma

(a) A

D E

Z B, C

(b)

Figure 1: (a) an example of Bayesian networks, (b) the tree structure as clustering B and C into Z [35]

Furthermore, if the Bayesian networks involve not only discrete random nodes but also decision nodes, then the BN2 can be extended into BN3 in the form of BN3= (DN, XN, L, P~

), where the decision node set, XN, is added.

In many domains, there may be continuous variables involved. In such circumstances, the continuous random nodes must be added into the Bayesian networks, which induces the fourth type of Bayesian networks BN4 in the form of BN4= (DN, CN, XN, L, P), where the continuous random node set CN is included,

Table 1: The Associated Conditional Probability Distribution of Figure 1(b) P(+a) = 0.20

P(+b|+a) = 0.80 P(+c|+a) = 0.20 P(+d|+b, +c) = 0.80 P(+d|+b, -c) = 0.80 P(+e|+c) = 0.80

P(+b|-y) = 0.20 P(+c|-a) = 0.05 P(+d|-b, +c)= 0.80 P(+d|-b, -c) = 0.05 P(+e|-c) = 0.60

Additionally, a general Bayesian network is normally acyclic. However, in some special situations, the Bayesian networks may be cyclic. The feedback loops in cyclic Bayesian networks imply the time-series dependency between the network nodes, which consequently expend the static Bayesian networks into dynamic Bayesian networks [4].

After the Bayesian networks are constructed as the knowledge bases, the decision makers need to reason from the knowledge bases. This kind of reasoning tasks is called abductive reasoning. The general form of abductive reasoning is explained in the following.

Remark 1 Abductive reasoning.

Given a set of evidence or observations Ĕ from a GBN, define the set of unknown nodes Û GBN\ Ĕ, the query of the belief (posterior) distribution of Û, BEL(Û| Ĕ), is an abductive reasoning problem. □

⊂

Since the conventional methods only answer very narrow scope of the queries on Bayesian networks, this dissertation develops several models to handle a set of specific reasoning problems in general Bayesian networks. In addition, these models are extended to consider the diagnosis and decision-making as well. Based on the four types of Bayesian networks

introduced previously, there are four categories of reasoning problems discussed in this dissertation:

1. Problem 1: diagnosis with discrete random nodes and crisp parameters. This category is reasoning from the simplest type of the Bayesian networks, BN1= (DN, L, P), and has been vastly studied in the literatures (Chapter 3).

2. Problem 2: diagnosis with discrete random nodes and fuzzy parameters in a static Bayesian network. This kind of problems is reasoning from BN2= (DN, L, P~

) (Chapter 3).

3. Problem 3: diagnosis and decision-making with discrete random nodes and fuzzy parameters in a static influence diagram. This kind of problems is solved on BN3= (DN, XN, L, P~

) (Chapter 4)

4. Problem 4: diagnosis and decision-making with continuous random nodes, decision nodes, and crisp parameters in a dynamic influence diagram. This type of problems is answered from BN4= (DN, CN, XN, L, P) (Chapter 5).

For every category of problems, this dissertation first gives a description of problem formulation, and develops the reasoning model in a comprehensive and systematic way.

Thereafter, the algorithms and solutions will be designed. One example or examples will be used to illustrate how to operate the reasoning methods, especially in medical informatics and supply chain systems. The outcomes and performances are examined carefully in the discussions. In the final chapter, some concluding remarks will be presented. The conceptual research framework and the dissertation structure are shown in Figure 2.

Introduction

Expert Systems and Probabilistic Reasoning

Problem 2: diagnosis with fuzzy parameters on BN₂= (DN, L, )

Problem 3: diagnosis and decision with fuzzy parameters on

BN₃= (DN, XN, L, )

Problem 4: diagnosis and decision with fuzzy parameters on

BN₄= (DN, CN, XN, L, P)

Discussions and Concluding remarks Bayesian networks and

Influence diagrams Fuzzy sets and theory

General Bayesian networks : GBN= (DN, CN, XN, L, P) Problem 1: diagnosis on Simplest BN: BN₁= (DN, L, P).

Chapter 1

Chapter 2

Chapter 3 Chapter 4

Chapter 6

Chapter 5

P~ P~

Figure 2: Research framework of the dissertation