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Modelling Belief Change in a Population Using Explanatory Coherence Bruce EdmondsCentre for Policy ModellingManchester 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) The idea of the presented model is to add a social contagion process to this Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 2
Adding Social Influence The idea is that a belief may be adopted by an actor from another with whom they are connected, if by doing so it increases the coherency of their set of beliefs Thus the adoption process depends on the current belief set of the receiving agent Belief revision here is done in a similar basis, beliefs are dropped depending on whether this increases internal coherence Opinions can be recovered in a number of ways, e.g. a weighted sum of belief presence or the change in coherence OR the change in coherence in the presence of a probe belief Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 3
Model Basics Fixed network of nodes and arcs There are, n, different beliefs {A, B, ....} circulating Each node, i,  has a (possibly empty) set of these “beliefs” that it holds There is a fixed “coherency”functionfrom 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 result in Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 4
Processes Each iteration the following occurs: Copying:  each arc is selected; a belief at the source randomly selected; then copied to destination with a probabilitylinearly proportional to the change in coherency it would cause  Dropping: each node is selected; a random belief is selected and then dropped with a probabilitylinearly proportional to the change in coherency it would cause -11 change has probability of 1 1-1 change has probability of 0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 5
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 AB 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 6
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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 7
Example – the randomly assigned coherency function just specified ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 8
5 different coherency functions Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 9
“Density” of A for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 10
“Density” of C for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 11
Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn Number of Beliefs Disappeared by time 500 Network Size Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 12
Av. Av. Resultant Opinion Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 13
Av. Consensus, Each Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 14
Zero Function ABC 0 AB BC AC 0 0 0 A B C 0 0 0  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 15
Consensus – Zero Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 16
Av. Resultant Opinion – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 17
The Fixed Random Fn ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 18
Consensus – Fixed Random Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 19
Single Function ABC -1 AB BC AC -0.5 -0.5 -0.5 A B C 1 1 1  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 20
Consensus – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 21
Av. Resultant Opinion – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 22
Prevalence of Belief Sets Example – Single Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 23
Double Function ABC -1 AB BC AC 1 1 1 A B C 0 0 0  -1 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 24
Consensus – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 25
Prevalence of Belief Sets Example – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 26
Comparing with Evidence Lack of available cross-sectional AND longitudinal opinion studies in groups But it can be compared with 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! Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 27
The End   Bruce Edmonds http://bruce.edmonds.name Centre for Policy Modelling http://cfpm.org These slides have been uploaded to http://slideshare.com

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Modelling Belief Change in a Population Using Explanatory Coherence

  • 1. Modelling Belief Change in a Population Using Explanatory Coherence Bruce EdmondsCentre for Policy ModellingManchester Metropolitan University
  • 2. 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) The idea of the presented model is to add a social contagion process to this Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 2
  • 3. Adding Social Influence The idea is that a belief may be adopted by an actor from another with whom they are connected, if by doing so it increases the coherency of their set of beliefs Thus the adoption process depends on the current belief set of the receiving agent Belief revision here is done in a similar basis, beliefs are dropped depending on whether this increases internal coherence Opinions can be recovered in a number of ways, e.g. a weighted sum of belief presence or the change in coherence OR the change in coherence in the presence of a probe belief Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 3
  • 4. Model Basics Fixed network of nodes and arcs There are, n, different beliefs {A, B, ....} circulating Each node, i, has a (possibly empty) set of these “beliefs” that it holds There is a fixed “coherency”functionfrom 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 result in Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 4
  • 5. Processes Each iteration the following occurs: Copying: each arc is selected; a belief at the source randomly selected; then copied to destination with a probabilitylinearly proportional to the change in coherency it would cause Dropping: each node is selected; a random belief is selected and then dropped with a probabilitylinearly proportional to the change in coherency it would cause -11 change has probability of 1 1-1 change has probability of 0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 5
  • 6. 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 AB 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 6
  • 7. 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 7
  • 8. Example – the randomly assigned coherency function just specified ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 8
  • 9. 5 different coherency functions Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 9
  • 10. “Density” of A for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 10
  • 11. “Density” of C for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 11
  • 12. Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn Number of Beliefs Disappeared by time 500 Network Size Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 12
  • 13. Av. Av. Resultant Opinion Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 13
  • 14. Av. Consensus, Each Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 14
  • 15. Zero Function ABC 0 AB BC AC 0 0 0 A B C 0 0 0  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 15
  • 16. Consensus – Zero Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 16
  • 17. Av. Resultant Opinion – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 17
  • 18. The Fixed Random Fn ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 18
  • 19. Consensus – Fixed Random Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 19
  • 20. Single Function ABC -1 AB BC AC -0.5 -0.5 -0.5 A B C 1 1 1  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 20
  • 21. Consensus – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 21
  • 22. Av. Resultant Opinion – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 22
  • 23. Prevalence of Belief Sets Example – Single Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 23
  • 24. Double Function ABC -1 AB BC AC 1 1 1 A B C 0 0 0  -1 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 24
  • 25. Consensus – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 25
  • 26. Prevalence of Belief Sets Example – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 26
  • 27. Comparing with Evidence Lack of available cross-sectional AND longitudinal opinion studies in groups But it can be compared with 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! Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 27
  • 28. The End Bruce Edmonds http://bruce.edmonds.name Centre for Policy Modelling http://cfpm.org These slides have been uploaded to http://slideshare.com