A Cooperative Coevolutionary Genetic Algorithm for Learning Bayesian Network Structures

Arthur Carvalho

a3carval@cs.uwaterloo.ca

Outline

• Bayesian Networks• CCGA• Experiments• Conclusion

Bayesian Networks

• AI technique– Diagnosis, predictions, modelling knowledge

• Graphical model– Represents a joint distribution over a set of

random variables– Exploits conditional independence– Concise, natural representation

Bayesian Networks

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Bayesian Networks

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Bayesian Networks

• Directed acyclic graph (DAG) – Nodes: random variables – Edges: direct influence of one variable on

another

• Each node is associated with a conditional probability distribution (CPD)

Bayesian Networks

• Learning the structure of the network (DAG)– Structure learning problem

• Learning parameters that define the CPDs – Parameter estimation– Maximum Likelihood estimation

Bayesian Networks

• Structure learning problem in fully observable datasets – Find a DAG that maximizes P(DAG | Data)– [Cooper & Herskovits, 92]

Bayesian Networks

• NP-Hard [Chickering et al, 1994]– The number of possible structures is

superexponential in the number of nodes [Robinson, 1977]

– For a network with n nodes, the number of different structures is:

Outline

• Bayesian Networks• CCGA• Experiments• Conclusion

CCGA

• Structure learning task can be decomposed into two dependent subtasks– To find an optimal ordering of the nodes – To find an optimal connectivity matrix

CCGA

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CCGA

• Two subpopulations– Binary (edges)– Permutation (nodes)

• Cooperative Coevolutionary Genetic Algorithm (CCGA)– Each subpopulation is coevolve using a

canonical GA

CCGA

• Evaluating individual species– Each subpopulation member is combined with

both the best known individual and a random individual from the other subpopulation

– The fitness function is applied to the two resulting solutions

• The highest value is the fitness

• CCGA-2 [Potter & De Jong, 1994]

CCGA

Operator Binary Permutation

Selection Tournament selection

Crossover Two-point crossover Cycle crossover

Mutation Bit-flip mutation Swap mutation

Replacement Preserve the best solution

Outline

• Bayesian Networks• CCGA• Experiments• Conclusion

Experiments

• Setup– K2 algorithm [Cooper & Herskovits, 1992]– Alarm network

• 37 nodes and 46 edges– Insurance network

• 27 nodes and 52 edges– Three datasets

• 1000, 3000, and 5000 instances– 100 executions

Experiments

• Parameters:Parameter Value

Generations 250

Population size 100

Probability crossover 0.6

Probability mutation:binary population

1 / max # of edges

Probability mutation: permutation population 0.5

Experiments

• Alarm network

Dataset Algorithm AverageStandard Deviation p-value

Alarm 1000−11,569.02

CCGA −12,166.21 178.230.0067

K2 −12,226.24 161.33

Alarm 3000−33,759.28

CCGA −35,020.31 396.060.0125

K2 −35,138.41 341.31

Alarm 5000−55,575.11

CCGA −57,282.66 548.55< 0.0001

K2 −57,574.12 492.94

Experiments

• Insurance network

Dataset Algorithm AverageStandard Deviation p-value

Insurance 1000−15,397.00

CCGA −15,787.28 279.19< 0.0001

K2 −16,107.61 293.06

Insurance 3000−43,508.63

CCGA −45,142.37 722.67< 0.0001

K2 −45,778.39 721.08

Insurance 5000−72,183.51

CCGA −74,624.88 995.01< 0.0001

K2 −75,837.82 1,249.83

Outline

• Bayesian Networks• CCGA• Experiments• Conclusion

Conclusion

• New algorithm to solve the structure learning problem– Novel representation– Good performance

• Future work– Incomplete datasets– Graph-related problems

Thank you!

Source code and datasets available at:www.cs.uwaterloo.ca/~a3carval/softwares/CCGA_code.rar

Arthur Carvalho

a3carval@cs.uwaterloo.ca