Modelling Belief Change in a Population Using Explanatory Coherence
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Transcript of Modelling Belief Change in a Population Using Explanatory Coherence
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