modelling belief change in a population using explanatory coherence
DESCRIPTION
A simulation model is presented that represents belief change, based on Thagard’s theory of explanatory coherence, within a population of agents who are connected by a social network. In this model there are a fixed number of represented beliefs, each of which are either held or not by each agent. These beliefs are to different extents coherent with each other – this is modelled using a coherence function from possible sets of core beliefs to [-1,1]. The social influence is achieved through gaining of a belief from another agent across a social link. Beliefs can be lost by being dropped from an agent’s store. Both of these processes happen with a probability related to the change in coherence that would result in an agent’s belief store. A resulting measured “opinion” can be retrieved in a number of ways, here as a weighted sum of a pattern of the core beliefs – opinion is thus an outcome and not directly processed by agents. Results suggest that a reasonable rate of copy and drop processes and a well connected network are required to achieve consensus, but given that, the approach is effective at producing consensuses for many compatibility functions. However, there are some belief structures where this is difficult.TRANSCRIPT
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Modelling Belief Change in a Population Using Explanatory Coherence
Bruce EdmondsCentre for Policy Modelling
Manchester Metropolitan University
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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)
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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
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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
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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
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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
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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
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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
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“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
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“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
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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
mb
er
of B
elie
fs D
issa
pe
are
d b
y tim
e 5
00
Nextwork Size
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Av. Av. Resultant Opinion
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 12
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Av. Consensus, Each Function
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 13
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Zero Function
A B C
ABC
AB BCAC
0
0 0 0
0
0 00
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Consensus – Zero Fn
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 15
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Av. Resultant Opinion – Fixed Random Fn
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 16
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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
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Consensus – Fixed Random Function
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 18
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Single Function
A B C
ABC
AB BCAC
0
1 1 1
-1
-0.5 -0.5-0.5
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Consensus – Single Fn
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 20
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Av. Resultant Opinion – Single Fn
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 21
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Prevalence of Belief Sets Example – Single
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 22
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Double Function
A B C
ABC
AB BCAC
-1
0 0 0
-1
1 11
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Consensus – Double Fn
Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 24
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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
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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!
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The End
Bruce Edmonds
http://bruce.edmonds.name
Centre for Policy Modelling
http://cfpm.org
These slides have been uploaded to http://slideshare.com