Cognitive Maps and Bayesian Networks

Emel Aktaş

Outline

Cognitive Maps

Influence Diagrams

Bayesian Networks

Motivation for Graphical Models

Systematic construction methods

Efficient inference procedures

Explicit encoding of independencies

Modular representation of probabilities

Taxonomy of Network Based Representation Schemes

Introduction

Person’s thinking about a problem or issue Tolman (1948)

Cognition

Various fields: Psychology Planning Geography Management

Some Definitions

Cause-effect networks Srinivas and Shekar (1997)

Graphical descriptions Axelrod (1976); Eden (1990)

Usage

Knowledge / belief representation

Capturing causality

Network-based formalisms Cognitive Maps (Axelrod, 1976) Belief Networks (Pearl, 1988) Qualitative Probabilistic Networks (Wellman, 1990)

Causal Maps

Causal map

Network representation of beliefs Nodes and arcs Directed graph

Harary et al. (1965); Harary (1972)

Ideas and actions Visible thinking

Mapping

Documentary coding

Interviews Subjective world of the interviewee

Questionnaire survey Groups

Personal construct theory Kelly (1955)

Representation

Short pieces of text

Unidirectional arrows

cause

Merging Decision of a College

Heads and tails

Incoming or outgoing arrows Negative relationships minus sign

Signed directed graph

Head: no outgoing arrows; goals / outcomes

Tail: no incoming arrows; options Centrality

Structural properties

Example(Ulengin, Topcu, Onsel, 2001)

Contribution to social

improvement Damage on the inhabitants of the surrounding area

by crossing

Damage to the historical texture of the

region

Cost of nationalization

Possibility of increasing

employment through creating jobs

-

Construction time

Facility in constructing (the topographic structure of

the crossing area and the surrounding land, etc)Suitability for

urban, regional and national

progress plans

-

Suitability for the transportation

policy

-

Financial damage in case of accidents during operation

headcentral

Problem/issue complexity

Cognitive scientists /organizational scientists

Central features total number of nodes total number of arrows cognitive centrality of particular nodes

Ratio of arrows to concepts 1.15 to 1.20 for maps elicited from interviews

The extent of the map

More nodes more complex

Mutual understanding of the issue

Number of concepts

length of the interview

skills of the interviewer

Representation

Graph See the causal relationships better

Matrix Mathematical analysis

Example: How can we motivate employees?

Variables Motivation Salaries Problems in the work environment Good attitude of the employer Good attitude of the colleagues Carreer possibilities

positive (+)

salary + motivation

Causal relationships between the variables

positive (+)

salary + motivation negative (-)

Problems in the work environment - motivation

Causal relationships between the variables

positive (+)

salary + motivation negative (-)

Problems in the work environment - motivation

No relationship (0)

attitude of colleagues 0 salary

Causal relationships between the variables

Determination of the causal relationships

Square matrix including all concepts Pairwise comparisons

mtv. sal. env. emp. col. car.

mtv. 0 0 0 0 0 0

sal. + 0 0 0 0 0

env. - 0 0 0 0 0

emp. + 0 - 0 0 +

col. + 0 0 0 0 0

car. + 0 0 0 0 0

How can we motivate employees?

mtv.

sal.

env.

+emp. +

-

col.

+

car.+

+

Causal Map of a Cement Producer

25

Economic setback

Investments on infrastructure,

residential and non-residential buildings

Development of

construction industry

Environmental concerns

Pressure on environmental

issues

Application of Kyoto Protocol

Competition

Regulations

Demand – supply balanceInput

costs

Capacity usage

Profitability

Cement demand

Restructuring of big players

Consolidation and vertical integration

+

+

+

+

+

+

-+

+-

-

-+

+

+

++

-

-+

+

Decreasing energy supply

+

Islands of themes

without accounting for hierarchy

Nodes in each cluster tightly linked

Bridges with other clusters minimized

Cluster Analysis

26

Hierarchical Clusters

Potent Options

Construction of the Group Cognitive Map

Gather related concepts from different persons Prepare a collective list of concepts Persons’ pairwise comparisons Construct of personal cognitive maps Aggregate personal cognitive maps

Single number of experts Taking experts’ opinions again about the doubtful relations

Size Over 100 nodes on the map

30

The most fundamental decisions are Definition of customer service (1) Forecasts of demand (8) Product routing (14) Information to be provided with the product (32)

The rest of the decisions cannot be taken unless these 4 decisions are given

Hierarchy of decisions

31

Definition of customer service

32

Centrality

33

First Cluster

34

Second Cluster

35

Influence Diagrams

Compact graphical /mathematical representation of a decision situation

It is a generalization of a Bayesian network, probabilistic inference problems

decision making problems

Influence Diagrams

37

Nodes Decision node [rectangle] Uncertainty node [oval] Deterministic node [double oval] Value node [diamond]

Arcs Functional arcs (ending in value node) Conditional arcs (ending in uncertainty /

deterministic node) Informational arcs (ending in decision node)

Influence Diagrams

38

ID of a Plan for Vacation

http://en.wikipedia.org/wiki/Influence_diagram

Decisions about the Marketing Budget and Product Price

http://www.lumina.com/software/influencediagrams.html

Bayesian Networks

Bayesian Networks

Probabilistic graphical model Variables

Probabilistic dependencies

Uncertain, ambiguous, and/or incomplete domains

Directed acyclic graphs

Bayesian Network Structure

A Bayesian Network has 3 components: X, S and P; X= {X1; X2;…; Xn} variablesS: causal structureP: conditional probabilities

Bayesian Network Structure

Rearrangement of the causal map Acyclic (no loops allowed) Direct and indirect relationships

Bayesian Network Structure

F is dependent on C and D A and B: root nodes F and G: leaves P(A,B,C,D,E,F,G)=P(G/D) P(F/C,D) P(E/B) P(D/A,B) P(C/A) P(A) P(B)

A B

C

F

D E

G

Bayesian Network Steps

1. Specification of the variables

2. Specification of the network structure

3. Determination of the conditional probabilities

4. Acquisition of additional knowledge

5. Inference based on knowledge

6. Interpretation of the results

Example: Product Development

4 variables Market Dynamics Product Life Cycle Market Leadership Rate of Product Launch

Dependence relations should be defined as conditional probabilities.

Nadkarni and Shenoy (2001)

Example: Product Development

Bayesian Network Structure

Bayesian Network Structure

P(D,C,L,R)=P(D)*P(C/D)*P(L)*P(R/C,L) Two variables are conditionally independent if

there is no arrow relating them No arrow between D and L L is

independent on D.

Knowledge Inference

Bayesian network is constructed Conditional probabilities are defined. It is possible to infer knowledge now using

specific software. Hugin (www.hugin.com) Netica (www.norsys.com)

Knowledge Inference

market dynamics

highlow

75.025.0

product life cycle

shortlong

73.726.3

market leadership

leaderfollower

90.010.0

rate of product launch

highlow

77.122.9

Knowledge Inference

market dynamics

highlow

0 100

product life cycle

shortlong

10.090.0

market leadership

leaderfollower

90.010.0

rate of product launch

highlow

52.048.1

A cognitive map – bayesian network application

Domain: tomography section within the radiology department of a private hospital in Turkey,

Objective: improve management system performance

The hospital operates 42 branches, including clinical research, diagnostics, and outpatient and inpatient care, with 279 expert physicians and 1038 healthcare and support staff.

A total of 36,000 radiological tests are conducted per annum.

Framework

Variables of the System

Preliminary Causal Map

Revised Causal Map

Discretization of the Variables

Final Causal Map

Compiled Bayesian Network

Additional Knowledge: Type of Scrutiny Known

The case where type of scrutiny and medicine treatment is known

Target values for the parent variables of time spent for scrutiny

Sensitivity of ‘‘time spent for scrutiny’’ based on findings at another node

References Axelrod, R., 1976. Structure of Decision. University of Princeton Press,

Princeton. Eden, C., 1988. Cognitive mapping: A review. European Journal of Operational

Research 36, 1-13. Harary, F., Norman, R., Cartwright, D., 1965. Structural Models: An Introduction

to the Theory of Directed Graphs. Wiley, New York. Harary, F., 1972. Graph Theory. Addison-Wesley, Reading. Pearl, J., 1988. Probabilistic Reasoning in Intelligent Systems: Networks of

Plausible Inference. Morgan Kaufmann, San Mateo, CA. Nadkarni S., Shenoy, P., 2001. A Bayesian network approach to making

inferences in causal maps, European Journal of Operational Research 128(3),479-498.

Srinivas V., Shekar B., 1997. Applications of uncertainty-based mental models in organizational learning: A case study in the Indian automobile industry.  Accounting, Management and Information Technologies, 7(2), 87-112.

Tolman E. C., 1948. Cognitive Maps in Rats and Man. Psychological Review 55: 189-208.

Wellman M.P., 1990. Fundamental Concepts of Qualitative Probabilistic Networks. Artificial Intelligence, 44(3):257–303.