Modelling Belief Change in a Population Using Explanatory Coherence

Bruce EdmondsCentre for Policy Modelling

Manchester Metropolitan University

Explanatory Coherence

• Thagard (1989)• A network in which beliefs are nodes, with

different relationships (the arcs) of consonance and dissonance between them

• Leading to a selection of a belief set with more internal coherency (according to the dissonance and consonance relations)

• Can be seen as an internal fitness function on the belief set (but its very possible that individuals have different functions)

Model Basics

• Fixed network of nodes and arcs• There are, n, different beliefs {A, B, ....}• Each node, i, has a (possibly empty) set of

“beliefs” that it holds• There is a fixed “coherency” function, Cn,

from possible sets of beliefs to {-1, 1}• Beliefs are randomly initialised at the start• Beliefs are copied along links or dropped by

nodes according to the change in coherency that these actions result in

Processes

Each iteration the following occurs:• Copying: each arc is selected; a belief at

the source randomly selected; then copied to destination with a probability related to the change in coherency it would cause

• Dropping: each node is selected; a random belief is selected and then dropped with a probability related to the change in coherency it would cause

Coherency Function

• Not just binary consistency/inconsistency but a range of values in between too (hence name)

• Could be mapped onto individuals’ reports of (in)coherence between beliefs

• Can allow a mapping from a formal logic to a coherency function so that model dynamics roughly matches reasonable belief revision

• Thus if we know AB and B↔C then Cn might be constrained by Cn({A, B})≥Cn({A}) and Cn({B, C})<0...

• ...so if there are any B’s around then a node with {A} in its belief set will likely to become {A, B} and a node with {B,C} will probably drop one of B or C

Example of the use of the coherency function• coherency({}) = -0.65• coherency({A}) = -0.81• coherency({A, B}) = -0.37• coherency({A, B, C}) = -0.54• coherency({A, C}) = 0.75• coherency({B}) = 0.19• coherency({B, C}) = 0.87• coherency({C}) = -0.56• A copy of a “C” making {A, B} change to {A, B, C} would

cause a change in coherence of (-0.37--0.54 = 0.17)• Dropping the “A” from {A, C} causes a change of -1.31

Example – the randomly assigned coherency function just specified

A B C

ABC

AB BCAC

-0.65

-0.81 0.19 -0.56

-0.54

-0.37 0.870.75

5 different coherency functions

Fn {} {A} {B} {C} {A,B} {B,C} {A,C} {A,B,C}

zero 0 0 0 0 0 0 0 0

fixedrand

.65 -.81 .19 -.56 -.37 .87 .75 -.54

sing 0 1 1 1 -1/2 -1/2 -1/2 -1

dble -1 0 0 0 1 1 1 -1

“Density” of A for different sized networks – Fixed Random Fn

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 80 155 230 305 380 455

5

10

15

20

25

“Density” of C for different sized networks – Fixed Random Fn

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 80 155 230 305 380 455

5

10

15

20

25

Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn

0

0.5

1

1.5

2

2.5

3

5 10 15 20 25 30 35 40 45 50Nu

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Nextwork Size

Av. Av. Resultant Opinion

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 12

Av. Consensus, Each Function

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 13

Zero Function

A B C

ABC

AB BCAC

0

0 0 0

0

0 00

Consensus – Zero Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 15

Av. Resultant Opinion – Fixed Random Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 16

The Fixed Random Fn

A B C

ABC

AB BCAC

-0.65

-0.81 0.19 -0.56

-0.54

-0.37 0.870.75

Consensus – Fixed Random Function

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 18

Single Function

A B C

ABC

AB BCAC

0

1 1 1

-1

-0.5 -0.5-0.5

Consensus – Single Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 20

Av. Resultant Opinion – Single Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 21

Prevalence of Belief Sets Example – Single

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 22

Double Function

A B C

ABC

AB BCAC

-1

0 0 0

-1

1 11

Consensus – Double Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 24

Prevalence of Belief Sets Example – Double Fn

Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 25

Comparing with Evidence

• Lack of available cross-sectional AND longitudinal opinion studies in groups

• But it might be possible to compare broad hypotheses– Consensus only appears in small groups (balance of

beliefs in bigger ones)– Big steps towards agreement appears due to the

disappearance of beliefs– (Mostly) network structure does not matter– Relative coherency of beliefs matters– Different outcomes can result depending on what gets

dropped (in small groups)• Ability to capture polarisation? To do!

The End

Bruce Edmonds

http://bruce.edmonds.name

Centre for Policy Modelling

http://cfpm.org

These slides have been uploaded to http://slideshare.com