Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian

Network

Vasilios Zarikas

1,2

, Elpiniki Papageorgiou

, Damira Pernebayeva

and Nurislam Tursynbek

School of Engineering, University of Apllied Sciences at Central Greece (TEI of Central Greece), Lamia, Greece

School of Engineering, Nazarbayev University, Astana, Kazakhstan

Keywords: Bayesian Networks, Decision Support System, Expert Systems, Fuzzy Rules, Medical Statistics.

Abstract: The task of carrying out an effective and efficient decision on medical domain is a complex one, since a lot

of uncertainty and vagueness is involved. Fuzzy logic and probabilistic methods for handling uncertain and

imprecise data both provide an advance towards the goal of constructing an intelligent decision support

system (DSS) for medical diagnosis and therapy. This work reports on a successfully developed DSS

concerning pneumonia disease. A detailed and clear description of the reasoning behind the core decision

making module of the DSS, is included, depicting the proposed methodological issues. The results have

shown that the suggested methodology for constructing bayesian networks (BNs) from fuzzy rules gives a

front-end decision about the severity of pulmonary infections, providing similar results to those obtained

with physicians’ intuition.

1 INTRODUCTION

Many techniques in the field of artificial intelligence

have been used to represent knowledge: production

rules, semantic nets, Bayesian nets, frameworks,

scripts, statements, logic, causal networks, among

others. Two significant topics of artificial

intelligence are fuzzy logic and bayesian probability

networks (Berner, 2007), (Konar and Chakraborty,

2005), (Konar, 2001). They have been shown to be

effective in the medical decision tasks (Pearl, 2005),

(Adlassnig, 1998), (Steimann and Adlassnig, 2000),

(Chen et al. 2005), (Sittig et al., 2008), (Hudson,

2006), (Fox et al., 2010), (Charitos et al., 2009),

(Fine et al. 1997). The choice of one of these two

techniques is based on two main factors: the nature

of the application and the designer’s skills. Both

decision making methods have been used in many

applications in medicine.

In the last decade, probabilistic reasoning and

fuzzy logic based methodologies were utilized in

handling imprecise data in pulmonary infections

(Pereira and Escuder, 1998), (Schurink et al., 2005),

(Aronsky and Haug, 1999), (Hoare and Lim, 2006),

(Saraoğlu and Sanli, 2007), (Cooper et al., 2005).

In this work, a useful step by step presentation of

the design of an implemented DSS and its reasoning

is given. It concerns pulmonary infections and a

decision making concerning the severity of the

disease (Zarikas et al., 2015). Physicians (stand as

medical experts) reported certain and uncertain

scientific knowledge concerning the disease of

pneumonia (Mani, 2000). The physicians expressed

their knowledge in the form of if-then rules. The

designer of the network in cooperation with the

experts/physicians assigned linguistic fuzzy values

to describe the probability between the observables

and the decision. Then these linguistic values were

transferred to numerical values using defuzzification

process in order to fill the conditional probability

tables. Finally, the system forecasts the severity of

pneumonia and drive a decision concerning their

admission in Internal Care Unit (ICU). The

simulations for test patients performed using the

implementation of the proposed methodology.

The main objective of this paper is to introduce,

analyze, and illustrate in a pedagogical way the

methodology that have already been described

mathematically in (Zarikas et al., 2015). Many

researchers contacted us and required a more

detailed description of the reasoning behind the

formulas appeared in (Zarikas et al., 2015). Because

of the relative novel character of the application in

the field of medical sciences, this paper gives a

detailed explanation on the proposed methodology

and the application preview the effectiveness of the

method.

Zarikas, V., Papageorgiou, E., Pernebayeva, D. and Tursynbek, N.

Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian Network.

DOI: 10.5220/0006642705390549

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 2, pages 539-549

ISBN: 978-989-758-275-2

539

The paper is organized into the following

sections: the second section presents a description on

Bayesian networks and influence diagrams. It also

provides a description on how fuzzy rules assigned

by medical doctors, are used to construct conditional

probability tables. The third section presents a

statement of the problem and how the BNs for the

specific problem is constructed by fuzzy rules

introduction. The fourth section provides a

description on stages of the development of the

correct topology used in the BN tool, presenting the

inference approach too. The fifth section outlines the

results and the main conclusions of the study.

2 BAYESIAN NETWORKS AND

FUZZY RULES

The definition of a consistent mathematical

framework that allows the integration of certain and

uncertain pieces of information into a plan of

reasoning, would provide a necessary knowledge

representation platform for every domain expert.

Such a model of knowledge representation already

exists and is known as belief network or Bayesian

Network (BN) or causal graph (Jensen, 2000),

(Pearl, 1988), (Stutz an Cheeseman, 1994),

(Friedman and Goldszmidt, 1998), (Heckerman and

Geiger, 1994).

Designing a Bayesian network means the

following tasks (i) define arcs from cause variables

to their effects; causal relationships reveal the

conditional dependencies and independencies, (ii)

assign values in Conditional Probability Tables

(CPT) based on prior knowledge and data, (iii)

finally appropriate algorithms have to been

employed (Pearl, 1986), (Pearl, 1987), (Pearl and

Verma, 1987) to determine various probabilities

from the network.

The synthesis of Utility theory and Bayesian

graph theory formulates the Decision theory

(Winkler and Robert, 1972), (Horvitz, 1988),

(Morgan and Bruce, 1968). The decision system that

is described in the present work follows the usual

assumptions. First we work with a set of mutual

exclusive actions and non-intervening actions i,e,

actions that their state is not correlated with P(H),

where H is the determining variable that affects the

decision. The expressive power of BNs becomes

obvious considering that they can encapsulate

statistical results, probability distributions, certain or

uncertain opinions, utilities, preferences, strategies,

goals and actions.

The fuzzy logic is based on fuzzy if-then rules

which have the general form “IF X is A THEN Y is

B,” where A and B are fuzzy sets. A fuzzy set is a

set containing elements that have varying degrees of

membership in the set. Elements in a fuzzy set,

because their membership need not be complete, can

also be members of other fuzzy set on the same

universe.

The physicians express their knowledge in the

form of fuzzy if-then rules due to the human

thinking approach. The experts accompanied with

the physicians assign linguistic fuzzy values

produced by each IF-THEN rule, to describe the

probability between the decision and the

observables. These linguistic values, through the

defuzzification approach of fuzzy logic, are

transferred to numerical values in order to fill the

conditional probability tables.

In order to show how the probability tables for

BNs are developed using the above type of if-then

rules, a generic approach is provided. Let’s consider

the following rule for the assessment of risk or

severity of an infection X: “IF symptom/observable-

A increases Then severity of infection X decreases”

(rule 1). This rule suggests information capable to

provide probabilities for the conditional probability

table (CPT) between the severity of infection and the

observable A.

The above rule is translated as: there is a

negative effect on severity from symptom A. This

means that the lower state of severity conditioned on

the higher state of symptom has a very very large

probability. In the simple case that both

``symptom/observable A” and ``severity of

infection/decision X” have only two states CPT

assignment is shown in Table 1. The inference of the

rule 1 could be described as:Rule Infer: The

probability P(severity-|A+)=very very high. The

linguistic description "very very high" might be

assigned with a fuzzy set with corresponding

membership functions.

Table 1: CPT of symbol A.

Symptom A

Severity X +-

+ complement 0.5

- v.v. high 0.5

The membership functions that constitute the

fuzzy sets which describe the inference of the fuzzy

rules are depicted in Figure1. This means that from

rule 1, there is a fuzzy belief which is assigned by

the fuzzy set shown in Fig. 1. After defuzzification

with the Center of Area method, a numerical value

of each fuzzy set is produced. The produced

numerical value is used to fill the probabilities in

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

540

CPT (Zarikas et al., 2015).

Therefore, we attempt to fill up the CPT for the

different states of A and severity of infection X

using this reasoning. In this point it worth stressing

that experts together with the physicians assign the

fuzzy values to describe the probabilities in CPT

between the decision and symptoms. First, they

extract the correct inference of the rule 1 i.e. that the

probability P(severity-|A+) is "very very high".

Second, the "very very high" probability is

transferred to the numerical value of 0.9, according

to the defuzzification process of the related fuzzy

sets. If there is no information for the effect on

severity X in the case that A decreases then neutral

policy is followed. This means that a probability of

0.5 for both ‘+’ and ‘–’ states of a variable is

assigned. This means that we assign probabilities

following neutral policy. Thus, Table 2 changes to

Table 3.

Table 2: CPT-probability (severity (X|A)).

Symptom A

Severity X + -

+ 0.1 0.5

- 0.9 0.5

Let's consider the “opposite” case of a rule in the

form: If symptom/observable A decreases Then the

severity of infection X increases. Now the Infer of

the rule is:

Probability P(severity+|A-)=v.very high and the

CPT table is completed in an analogous way

resulting to probability P(severity+|A-)=0.9. It is

obvious that A is in general a different symptom

than the one mentioned before in Tables 1, 2.

There are also cases that the number of states of

the severity of a disease is more than two. In what

follows, an explanation of how it is possible to

construct CPTs for such multistate variables based

on fuzzy rules is given. Let us work with the rule 1,

“If severity A increases then severity of infection X

decreases”. Assuming that the symptom/observable

A is described by three states: {weak, moderate,

strong} and the severity or risk of infection X has

four states: {small, medium, high and very high},

then for the CPT it is needed to assign values for:

P(very small severity|A weak), P(very small

severity|A strong), P(very small severity|A

moderate), P(small severity|A weak), P(small

severity|A strong ), P(small severity|A

moderate),....etc.

These probabilities are proposed to be described

by eight (8) membership functions for

P(severity_state

|A moderate) and

P(severity_state

|A strong ) borrowed from fuzzy

logic methodology. Finally all P(severity_State

weak) are equal to 1/4 due to neutral assignment

policy. These membership functions have been

defined by the related fuzzy sets as illustrated in

Figure 1.

Figure 1: Membership functions used.

Thus the probability to decrease severity X as A

increases could be assigned in a numerical value 0.7

derived by fuzzy sets as presented in membership

functions describing the “high” probability. The

following CPT for the different states of A and

severity X is needed to fill up Table 3 considering

the above fuzzy sets and their ranges.

Table 3: Probability (severity (X|A)) for multistate

example.

Symptom-A(state)

Severity X Weak Moderate Strong

small - medium high 0.6 High 0.7

med -

weak

0.3

very weak

0.2

high - very weak 0.2 v.v.wea

0.1

very high - 0 0

Next, Table 4 is filled up respecting axioms of

bayesian probabilistic theory. Furthermore, neutral

policy was also applied for the entries we have no

information coming from the rule. It is worth

mentioning that if a companion rule of the form “If

symptom/observable A decreases Then the severity

of infection X increases” then it would be possible to

fill all the entries of the CPT.

Table 4: Probability (severity (X|A)) for multistate

example.

Symptom A

Severity-X Weak moderate strong

Small 0.25 0.55 0.7

Med 0.25 0.3 0.2

High 0.25 0.15 0.1

Very high 0.25 0 0

Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian Network

541

Let us now consider another type of medical rule

with two observables A and B to determine the

severity of an infection X. The severity of the

infection X is considered to have four states: {small,

medium, large, v.large}. The physician assigns next

rule to determine severity:

``IF observable A is "Yes" and observable B

decreases THEN severity of infection is medium”.

This rule could be infer the probabilities:

P(Severity med|low, Yes)=v.high (equals to 0.8)

P(Severity med|moderate, Yes)=high (equals to 0.7).

The CPT for the different states of observables A

and B and severity of infection X is filled up as in

Table 5.

Table 5: CPT-Probability (severity |AB,BA).

(A)NO (A)YES (exist)

(B)

low

(B)

mod

(B)

high

(B)

low

(B)

mod

(B)

high

Small

Medium 0.8 0.7

Large

v. large

Next step is to normalise the columns (A)YES-

(B)low and (A)YES-(B)mod and finally complete all

the other collumns following neutral policy. Thus,

Table 6 (CPT-completed filled-

Probability(Severity|B,A)) is derived.

Table 6: CPT-completed Probability (severity |B,A).

Observable A

Severity NO YES (exist)

low mod high low mod high

Small 0.25 0.25 0.25 0.1 0.15 0.25

Medium 0.25 0.25 0.25 0.8 0.7 0.25

Large 0.25 0.25 0.25 0.1 0.15 0.25

v. large 0.25 0.25 0.25 0 0 0.25

3 PROBLEM AND TARGET

Some common criticisms about applied Bayesian

networks concern the necessity of filling correctly a

lot of conditional probability tables. However, the

involvement of all these probability tables, is the

reason that makes this decision tool extremely

precise, expressive and mathematically consistent.

BNs indicate emphatically to any decision builder

how many pieces of information are involved for a

precise decision making. The required big set of

probabilities by no means can be disregarded

unwisely for the sake of simplicity or approximation

or a fault decision will be driven. However, it is

possible to find methods for filling in a correct way

the missing pieces of information. The present work

describes such a method for the problem under

study.

Another issue is that experts complain that the

human brain does not work in this way and even

scientists (not experienced in “Bayesian language”)

cannot easily report safely all these numbers in order

to describe a domain knowledge. A practical

solution of this problem is presented in this work for

a particular medical case. However, the selected

medical decision problem is not a special one but a

quite typical and general case. Experts report a list

of rules containing estimates about probabilities.

These rules are a subset of all the possible rules that

the full problem would require and the reasoning

and the justification behind this reduction is

explained in the relevant sections below.

For the chosen medical problem of pneumonia

(pulmonary infections) the prediction of severity is a

complicated process with many parameters, factors

and preconditions (Gennis et al., 1989), (Langer

1994). See also CDC Criteria for Defining

Nosocomial Pneumonia, online available in http://

www.cdc.gov/.

For the problem of pneumonia, a number of

typical symptoms are associated. If pneumonia is

suspected on the basis of a patient's symptoms and

findings from physical examination this indicates

that more tests are needed to confirm the diagnosis.

The set of all these data provide a basis for

evaluation the severity of infection and the need for

intensive care (Schurink et al., 2005).

Thus, severity of getting infected by pneumonia

can be approximated by observing several

symptoms. In the present work, three physicians

(stated as experts), from the General Hospital of

Lamia, Greece, were selected at first to define the

number and type of symptoms-observables affecting

the problem of pulmonary infection. Thirty-four

different symptoms were reported, named from C1

to C34. These symptoms listed in Table 7, are well

documented in bibliography. These are the main

variables that have an important role in the final

diagnostic inference. For this application,

symptom/observable values take either two, three,

four or five possible discrete or fuzzy values, as

shown in Table 8. Each one variable/observable has

different states, for example C4 (fever) is separated

into five fuzzy values: no fever (36-38.4C), low

grade (38.5-38.9C), moderate, high grade,

hyperpyrexia (>41^{0}).

Next, the three physicians (expert doctors) were

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

542

interviewed in order to construct a certain list of

rules containing estimates of the probability of

infection. Such rules, defining which symptoms

increase or decrease the risk of infection, can build

the base for a Bayesian network. The target is to

encode the medical expert's knowledge about

pneumonia in a Bayesian network. The complete set

of rules can be found in the published work (Zarikas

et al., 2015) in Appendix, “Rules”. Rules have been

given in the form of:

If Cn {increases|decreases|exists}then the risk of

infection is {small|medium|large|very_large} or

{increases, decreases},

while for two or more symptoms in the following

form:

If Cn {increases|decreases|exists} and Cm

{increases|decreases|exists}then the risk of infection

is {small|medium|large|very_large}or {increases,

decreases},

where Cn and Cm are two different symptoms.

Experts have also stated that most of the times

doctors know evidence for one, two or three at most

symptoms from an examined patient.

Here it is worth pointing out that the proposed

decision module is not a rule based expert

subsystem. It is rather a probabilistic decision

subsystem encoding medical “rules” expressing

certain and uncertain knowledge. In order to design

and implement the full Bayesian network with all

the conditional probability tables a much larger set

of rules is needed to cover them. However

interviewing doctors, a medically correct strategy

was constructed in order to fill the gaps in the

probability tables.

So far, we have only presented a general

approach on how from the previously mentioned

fuzzy rules, CPTs are constructed. In what follows,

the proposed approach as well as the overall

reasoning is explored and analyzed to the particular

problem to accomplish the final decision.

Table 7: Concepts coding pulmonary infections.

Nodes

C1: Dyspnea

C17: Radiologic evidence of

complicated pneumonia

C2: Cough C18: Acidity (pH)

C3: Rigor/chills C19: Partial pressure of oxygen

C4: Fever C20: Partial pressure of CO2

C5: Loss of appetite C21: Oxygen saturation O2%

C6: Debility C22: White blood cells (WBC)

C7: Pleuritic pain C23: Immunocompromise

C8: Heamoptysis C24: Comorbidities

C9:Oxygen

requirement

C25: Age

C10: Tachypnea C26: Sputum culture

C11:Acoustic C27: Bronchial secrets culture

characte

C12:(Glascow Comma

Scale)

C28: Blood culture

C13:Systolic-Blood

Pressure (mmHg)

C29:Pleural Fluid culture

C14: Diastolic blooΙf C30: Mantoux

C15:Tachyca

dia C31:Gram stain (gram (+)

C16:Radiologic

pneumonia

C32: Urinary antigen test

C33: Pathogen Sensitivity

4 DESIGNING THE BAYESIAN

NETWORK

In this section we describe how a Bayesian network

is designed for the particular medical problem. We

are going to create three types of nodes:

• for every symptom a symptom node

• for every rule or group of paired rules a rule node

• for every rule node a severity utility node

• one central utility comprising the overall utility

and one decision node concerning admission to

the ICU or not.

4.1 Symptoms

As a first step the symptoms should be entered in the

BN, see Fig.2. In general, one symptom has different

number of states according to physicians-expert

knowledge and medical guidelines. Consequently,

these states are associated with fuzzy membership

functions, see Table 8 for examples.

Every symptom is represented by a probability

informational node called Cn with the symptom's

states as possible values for the node. From now on,

these nodes will be denoted as symptom nodes.

The probability table of these nodes can be filled

with certain or uncertain information (prior

probabilities/evidences). Physicians report their

knowledge by providing fuzzy rules based on their

knowledge and guidelines from which our system

extracts probabilities. This extraction is done

following the method described in the section 3.

Table 8: Examples for symptoms.

Symptom Type of values (discrete or fuzzy)

C7 Pleuritic

pain

Two discrete values: 0, 1

C4 Fever

Six Fuzzy values (“hypothermia” (34-36),

“no fever” (36-38.4), “low” (38.5-38.9),

“moderate” (38.9-39.5), “high” (39.5-

40.9), “hyperpyrexia” (>41))

C23: Immuno Two fuzzy values (presence, absence)

Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian Network

543

Figure 2: Symptom nodes.

Values according to the results of patient's

examinations or according to physicians's subjective

appraisal are entered. In case that there is no

evidence for one particular symtom statistical data

that can provide frequencies may be used either for

the group that the patient belongs to, or for the

patient's historic profile. Thus, for a particular

case/patient the probability table for the symptom

C7 Pleuritic pain could look like this opposed in

Table 9.

Table 9: Probability table for the sympton C7 Pleuritic

Plain.

C7 Pleuritic pain Probability

State0 0.3

State1 0.7

Thus, the value 0.7 for the probability of State1

can arise from statistics about a high-risk group or

from a doctor's judgment or patient’s examination,

see Table 9. In case a physician detects no clarity or

definiteness on the answer of a patient he can assign

a probability less than unity. If there is no evidence

about the prior probabilities of a symptom node then

a neutral policy regarding the prior probabilities can

be applied (assign equal probabilities for a multiple

state symptom). In most cases, soon after the

patient's visit to the medical center only a few

symptoms can be reported or measured with a

certain or uncertain degree of belief. For the rest of

them that remain unspecified, either a neutral policy

should be followed in order to assign prior

probabilities or (as in our case) the setup of an

algorithm which disregards from the whole BN all

the child of the non-relevant symptom nodes is

needed.

4.2 Implementing the Rules

Medical rules encode the necessary information with

the help of which, it is possible to associate

probabilistically disease’s severity and symptoms.

Table 10: Conditional probability table of the risk of

infection node as a direct children.

Symptom

State 0 …

Symptom

State 0 State 1

State

.. ..

Symptom 3

State

…

State

… .. . …

.. .. .. . .. . …

large

severit

p1 p2 p3 … … … … .. . …

small

severit

1-p1 1-p2 1-p3 … … … … .. . …

4.2.1 A First Topology

Naively, since all symptoms causally influence the

node severity, a connection of all symptom nodes to

the central node representing severity is expected.

However, this design results to an non solvable

topology. It generates a quite large probability table.

Furthermore, for adding a new rule, it would be

required to modify the values for the previous

entered rules! In summary, a CPT that represents

more than two or three rules is not manageable.

Table 10 illustrates clearly the complexity of the

CPT in the simple case of a two-state severity node.

The row large severity encapsulates the chances for

large severity of infection, given the states of the

column for the symptoms

1, 2

( ,..., )

, where

N is given by

Nn



with i=1,2,..k. where k is

the number of symptoms and

the number of

states of symptom

i. The row small severity

contains the complementary probabilities. It is

obvious that although one rule concerns only one or

two symptoms the topology results to the fact that

every rule affects all probabilities in the CPT.

Nevertheless, this is the correct topology that ideally

represents the modelling of knowledge for the

disease and its set of symptoms. This clearly shows

all the elementary pieces of knowledge involved.

Thus, in some cases, where a set of rules encode

critical information, it may be necessary to acquire

the relevant knowledge and construct a part of this

very detailed topology.

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

544

4.2.2 Tractable Topology

A new simplified topology that works very

efficiently is generated if for every rule a single rule

node is assigned (see Figure 3). Fr this reason we

name each rule node as sev-Cn or sev-Cn-Cm where

n and m identify the symptoms involved in the rule.

Rule nodes are causally affected by the symptom

modes. In general, one or more symptoms affect one

or more rules. Sometimes, it is preferable, two rules

with the same symptoms to be combined in one rule

node. Figure 3 shows the improved BN containing

all symptoms and all rule nodes. Each rule node can

be viewed now as a determining variable.

Figure 3: Rule nodes.

Although this is not a complicated topology

many conditional probabilities have to be

determined. Naively, one can say that the available

rules provide less than the required amount of

information. However, as we have previously

explained physicians interpret and work with them

in a way that allows to fill in the table. Let's explain

the proposed method with one more example based

on the following rule: “if symptom A is strongly in

state1 (one of three states) then the severity of

infection is large”. Apart from the obvious

information that the conditional probability of large

severity is a number close to unity conditioned on

state 1, we can deduce more information. Following

physicians' instructions, reported in the interviews,

for some particular rules a complement (in our case

small) probability for severity of infection can be

assumed if the condition of the rule isn't satisfied.

However, note that for most cases the negation

of the first part of the rule is connected with no

preference i.e. leads to neutral assignment of

probabilities for risk states. Such cases have already

been presented in the previous section. Now what

about the other states; if the states of the symptom A

comprise an ordinal scale (state "1" is smaller than

state "2" and state "2" smaller than state "3") then in

most cases, except if physicians state otherwise, it is

allowed to understand that the rule remains less true

for symptom in state “2” and not true for symptom

in state “3”.

Let's consider rule 30 and rule 31, see (Zarikas et

al., 2015). These two rules can be represented by

one combined rule node. When both these rules are

not satisfied it means that they point to a not large

severity. One can assign a conditional probability

equal to 1 for C22 on state “normal” and on state

“small” severity. Since the condition of the rule is

not satisfied (C22 is neither in the higher state nor in

the lower state) we assign 1 to the probability for a

small risk independently of what is the state of C1.

Alternatively another possible assignment is to set

probability equal to one for P(severity|normal,state0)

and P(risk|normal,state1) and a probability close to 1

(for example 0.9) for the other two

P(severity|normal,state2) and

P(severity|normal,state3) since the latter are

associated with states of increased C1-dyspnea.

Table 11 presents a first realization of rule 30 and its

companion rule 31:

Table 11: Conditional probability table of the risk of

infection node as a direct children.

C22 leuko

enia Normal Leukoc

tosis

C1 ...

State

...

small ... 1 1 0.8 0.8 ...

ediu

... 0 0 0.15 0.15 ...

lar

e ... 0 0 0.05 0.05 ...

very

lar

... 0 0 0 0 ...

Note that if in Table 11 all entries in the first raw

(small severity) are set to one while all entries in all

other rows (medium, large, very large) are set to

zero, no large modifications will be raised in the

final provision of decisions.

If C22 has the value leukocytosis means that C22

have increased and therefore the condition of the

rule 30 is satisfied, provided that C1 is increasing

too, pointing to large severity. Linear interpolation

provides values in between, for the row “large”, see

Table 12.

In case that C1 is in State1 or State2,

P(medium|leukocytosis,1or2) is essential to be

greater than 0. For example, it could be initially

P(medium|leukocytosis,1or2)=0.3 and

P(medium|leukocytosis,0or3)=0.

Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian Network

545

Table 12: First part of the probability table for rule C22-

C1.

C22 Leukopenia Normal Leukocytosis

C1 ... ...

State

small ... ... 0.7 0.5 0.2 0

Medium ... ... - - - -

large ... ... 0.2 0.5 0.8 1

Very large ... ... - - - -

Next, row “large” is kept the same while reduced

values of probabilities are set to the CPT entries

above and below a specific element of row “large”.

Consequently, every column is normalized to a sum

of 1 in each column. Therefore, following this

reasoning, we retune values in Table 12, composing

Table 13:

Table 13: First part of the CPT for rule C22-C1.

C22 Leukopenia Normal Leukocytosis

C1 ... ... State

State

small ... ... 0.5 0.1 0 0

medium ... ... 0.15 0.2 0.1 0

large ... ... 0.2 0.5 0.8 1

Very

large

... ... 0.15 0.2 0.1 0

Keeping the same reasoning (explained for

leukocytosis), the column leukopenia can be

determined by rule 31. The usage of all given riles

resulted to the development of 102 rule nodes in the

final decision network.

4.2.3 Connecting the Rule Nodes

After defining and setting all the rule nodes, the next

step connects them with a utility node and through it

to the final decision node. The Utility node can be

modelled like Table14 and associates the decision

node with the determining variable or variables. In

this table a determining variable representing the

total information which concerns the severity of the

infection has been assumed. The decision node has

been modeled with two states “Admission” meaning

ICU admission and “No admission”. Table 14

contains utility values for the four states of the

determining variable. It will become apparent below

that it is preferable to design more than one utility

nodes (one utility for a group of similar determining

variables). Now, determining variables are the so

named rule nodes.

Table 14: First part of the CPT for rule C22-C1.

Determining

variable of

severity

ICU admission

no ICU

admission

low 0 1000

medium 330 660

large 660 330

very_large 1000 0

The utility table is not a uniquely determined

quantity. It reflects the strategy of the domain expert

and thus different utility values can be set,

depending on how conservative or strict is the

selected policy. Physicians, suggest that a larger

certainty for one state of severity of infection should

be given, if more rules point towards it. However, in

some cases that evidence is given (updating the prior

probabilities of certain symptoms for a patient) a

discrepancy may arise. If one or more rules indicate

a small risk, while other rules indicate a large risk

the implemented software provides a warning signal.

This build in check of all active rules enhance

significantly the performance of our decision tool.

Note, that active rules are the rules that have been

activated by updating the prior probabilities of

certain symptoms.

4.2.4 Connecting All Rules

It was explained why it is not a wise choice to

connect all the 102 rule nodes to one central utility

node. The central utility node contains a double-row

table with utility weights expressing the strength of

infection for the given states of the parent nodes

(Table 15). Therefore, a column represents one

combination of states of the parents’ nodes. Every

rule node has four states small, medium, large and

very large. So for every possible assignment of each

of the 102 nodes with one of the 4 values, we need a

utility table with

102

columns, which are too many

for a decision system to deal with.

However, for clarification reasons further

elaboration on this topology with one utility will be

devoted in order to illustrate the meaning of the

entries of this utility table. The decision node ICU

admission with the states yes and no allows defining

utility values for these two states, see Figure 4 and

Table 15.

A problem that appears usually in medical

diagnostic decision systems is the possibility the

reported symptoms to lead to a serious discrepancy

regarding the risk/severity of infection. It is always

possible a patient to report mistakenly a symptom

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

546

that activate rule or rules that point to a very large

risk/severity of infection while other symptom or

symptoms point to small chance of infection.

Therefore, a separate check for a discrepancy by an

appropriate algorithm is necessary.

A significant remark is that in realistic cases only

few rules from the whole set are activated by the

updated symptoms. This has an important

consequence. All these rule nodes that have not been

activated by updated symptoms, are associated with

a parent symptom node with neutral set of

probabilities. This finally drives the system towards

a neutral decision. If most rules are not activated and

only two or three rules indicate with 0.8 chance a

large severity, the final result through the utility

node would point to a medium level expected utility

value in the scale of 0-1000. This fact provides

difficulties for the system to drive a clear positive

infection decision. However, this is also an issue that

can easily resolve with the help of a special

algorithm that excludes non-activated rules.

The proposed solution includes an extra layer of

utility nodes. Each rule node is connected to a utility

node i-Cn-Cm which is named “admission node”

(see figure 4). Thus, the role of the admission nodes

is to convert the probabilistic representation of the

severity of infection into a utility value as it can be

seen in Table 15.

The central utility (weighted severity for

admission) is the chιld of all admission nodes (see

figure 4). This final node combines the utilities on

admission from all rules together. Technically the

central utility node is expressed by a utility node of

type MAU (Multiple Algorithm Utility). MAU,

utility node integrates many utility nodes with the

help of a mathematical expression depending on the

values of their parent nodes. It uses typical functions

such as sum, division, maximum/minimum and

logical operations.

Table 15: Theoretical Central Utility Table.

Rule 1 small small ... small

Rule 2 small small ... large

... ... ... ... ...

Rule n small small ... medium

Infection positive negative ... positive

Value 50 950 ... 500

Rule 1 small ...

Very

large

Very

large

Rule 2 large ...

Very

large

Very

large

... ... ... ... ...

Rule n medium ...

Very

large

Very

large

Infection negative ... positive negative

Value 500 ... 1000 0

Figure 4: Inflection nodes (in blue).

The decision tool was developed based on the

code of Bayes Fusion, given for academic use This

software provides a built-in maximum function for

MAU nodes. The code cannot support an estimation

of the maximum over all rule values if they are more

than 20. One solution is to collect the infection

nodes into small groups and then calculate the

maximum of the maximums. Another way is to

assign equal weight to utilitiy of each rule.

In order to evaluate the severity of infection there

is no need to take under consideration all the set of

the rules. Since only a few symptoms are given for a

particular patient, the rules that contain these

symptoms will control the decision. Consequently, a

few rules are often updated with non neutral prior

probabilities which result in a noteworthy risk.

However, as we have noticed the contribution of all

the rest may affect considerably the final decision.

This problem is resolved easier with the extra layer

of utility nodes and the addition of a central node

with a maximum MAU nodes evaluation scheme.

As a last step it's necessary to ensure that there

are no discrepancies otherwise a notification has to

be provided. As mentioned earlier we expect every

symptom to report more or less the same results. If a

patient has a very large and a small value of severity

at the same time, something went wrong. The tool

have implement an algorithm which evaluates the

differences of the minimum and the maximum of all

risks given by the selected rules. If the difference is

above a predefined threshold the user is notified

about.

5 RESULTS AND DISCUSSION

After construction of Bayesian network using the

Bayes Fusion platform (https://www.

Medical Decision Support Tool from a Fuzzy-Rules Driven Bayesian Network

547

bayesfusion.com/), a number of patient cases have

been examined in order to set evidences to the

network and illustrate its decision-making

capabilities. Specifically, (84) decision making cases

on pneumonia severity assessment have been

derived from a randomly selected set of anonymous

patients with confirmed pneumonia. The decision-

making capabilities of the technique was presented

by simulating these patient cases and estimating the

outcomes. The results have been reported in (Zarikas

et al., 2015).

This work provides a pedagogical description of

all the methodology that was followed to design the

implemented DSS. It is a response to many requests

to provide a clear explanation of the reasoning

behind the formulas presented in (Zarikas et al.,

2015). First, a new methodology for construction of

BNs using if-then rules and main aspects of fuzzy

logic is clearly presented and second, the efficient

modeling and reasoning concerning the

implementation of all rules to a network with a

specific topology, is given. The method, we

presented in this paper can be generalized to similar

fuzzy rule bases.

Novel ideas that have been materialized in the

DSS are: 1) Physicians have not been involved for

the probability assignments but only for reporting

and explaining the rules 2) Fuzzy rules have been

translated into probabilities 2) There is an

intermediate layer of utilities that transfer their

values to a central utility node 4) The fuzzy rules are

comprehensive enough for a physician, and describe

a simple symptom/disease causal relation. A

particular set of patients with pulmonary infections

were studied as a first preliminary test of the

decision making system on severity assessment and

show the methodology's performance.

Future work is focused to analyze and implement

this approach in other domains and decision

problems, to include more knowledge and

information types for the decision model

enhancement. Specifically, extracted knowledge

from other sources except physicians’ suggestions,

such as data through data mining and medical

guidelines, will be taken under consideration for the

model enhancement.

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