A Genetic Algorithm for Learning Bayesian Network Adjacency Matrices from Data

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense


Benjamin B. Perry

Laboratory for Knowledge Discovery in Databases

Kansas State University


http://www.kddresearch.org

http://www.cis.ksu.edu/~bbp9857

A Genetic Algorithm for Learning

Bayesian Network Adjacency Matrices from Data

Ben Perry
–

M.S. Thesis Defense

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Bayesian Network

–
Definitions and examples

–
Inference and learning

•
Genetic Algorithms

•
Structure Learning Background

–
Problem

–
K2

algorithm

–
Sparse Candidate

•
Improving
K2
: Permutation Genetic Algorithm (GASLEAK)

–
Shortcoming: greedy, sensitive to ordering

–
Permutation GA

•
Master’s thesis: Adjacency Matrix GA (SLAM GA)

–
Rationale


•
Evaluation with Known Bayesian Networks

•
Summary

Overview

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Bayesian Network

–
Directed acyclic graph


–
Vertices

(nodes): denote events, or states of affairs (each a random variable)

–
Edges

(arcs, links): denote conditional dependencies, causalities

–
Model of
conditional dependence assertions

(or
CI assumptions
)

•
Example (“Ben’s Presentation” BBN) (sprinkler)







•
General
Product (Chain) Rule

for BBNs`

Bayesian Belief Networks (BBNS):

Definition

X
1

X
2

X
3

X
4

Sleep:

Narcoleptic

Well

Bad

All
-
nighter

Appearance:

Good
, Bad

Memory
: Elephant,
Good
, Bad, None

Ben is nervous:

Extremely, Yes,
No

X
5

Ben’s presentation:

Good
, Not so good, Failed miserably

P
(
Well
,
Good
,
Good
,
No
,
Good
) =
P
(
G
) ∙
P
(
G

|
W
) ∙
P
(
G

|
W
) ∙
P
(
N

|
G
,
G
) ∙
P
(
G

|
N
)

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Idea

–
Want: model that can be used to perform inference

–
Desired properties

•
Correlations among variables

•
Ability to represent functional, logical, stochastic relationships

•
Probability of certain events

•
Inference: Decision Support Problems

–
Diagnosis (medical, equipment)

–
Pattern recognition (image, speech)

–
Prediction

•
Want to Learn: Most Likely Model that
Generates

Observed Data

–
Under certain assumptions (
Causal Markovity)
, it has been shown that we can
do it

–
Given: data
D

(
tuples

or vectors containing observed values of variables)

–
Return: directed graph (
V
,
E
) expressing
target CPTs

–
NEXT:
Genetic algorithms

Graphical Models

of Probability Distributions

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Idea

–
Emulate natural process of survival of the fittest (Example: Roaches adapt)

–
Each generation has many diverse individuals

–
Each individual competes for the chance to survive

–
Most common approach: best individuals live to the next generation and mate

–
Produce children with traits from both parents

–
If parents are strong, children might be stronger

•
Major components (operators)

–
Fitness function

–
Chromosome manipulation

–
Cross
-
over (Not the “John Edward” type!), mutation

•
From (Educated?) Guess to Gold

–
Initial population typically random or not much better than random
–

bad scores

–
Performs well with a non
-
deceptive search space and good genetic operators

–
Ability to escape local optima with mutations.

–
Not guaranteed to get the best answer, but usually gets close

Genetic Algorithms

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Learning Structure:

K2

Algorithm

•
Algorithm
Learn
-
BBN
-
Structure
-
K2

(
D, Max
-
Parents
)

FOR
i



1瑯
n

DO


// arbitrary ordering of variables {
x
1
, x
2
, …, x
n
}

WHILE (
Parents
[
x
i
].
Size

<
Max
-
Parents
) DO

// find best candidate parent


Best



argmax
j>i

(
P
(
D

|
x
j



Parents
[
x
i
]
)

// max Dirichlet score


IF (
Parents
[
x
i
] +
Best
).
Score

>
Parents
[
x
i
].
Score
) THEN
Parents
[
x
i
] +=
Best

RETURN ({
Parents
[
x
i
] |
i



{1, 2, …,
n
}})

•
A

L
ogical
A
larm
R
eduction
M
echanism [Beinlich
et al
, 1989]

–
BBN model for patient monitoring in surgical anesthesia

–
Vertices (37): findings (e.g.,
esophageal intubation
), intermediates, observables

–
K2
: found BBN different in only 1 edge from
gold standard

(elicited from expert)

17

6

5

4

19

10

21

31

11

27

20

22

15

34

32

12

29

9

28

7

8

30

25

18

26

1

2

3

33

14

35

23

13

36

24

16

3
7

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Learning Structure:

K2

downfalls

•
Greedy (may fall into local maxima)

•
Highly dependent upon node ordering

•
Optimal node ordering must be given

•
If optimal order is already known, an expert could probably create the network

•
Number of orderings consistent with DAGs is exponential (n!)



Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
General Idea:

–
Inspect k
-
best parent candidates at a time. (K2 only inspects one)

–
k is typically very small ~ 5 ≤ k ≤ 15

–
Exponential to the order of k

•
Algorithm:

Loop until no improvements or iteration limit exceeds:


For each node, select the top k parent candidates (mutual information or
m_disc) [Restrict]


Build a network by manipulating parents (add, remove, reverse from
candidate set for each node) . Only accept changes that maximizes the
network score (Minimum Descriptor Length) [Maximize phase]

•
Must handle cycles.. expensive.

–
K2 gives this to us for free

–
Next: Improving K2

Learning Structure:

Sparse Candidate

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

GASLEAK
:

A Permutation GA for Variable Ordering

[2] Representation Evaluator

for Bayesian Network

Structure Learning Problems




















G
enetic
A
lgorithm for
S
tructure
L
earning

from
E
vidence,
A
IS, and
K
2

D
: Training Data


: Evidence Specification

D
train

(Structure Learning)

D
val

(Inference)

[1] Permutation Genetic Algorithm

α


Candidate

Ordering

f
(
α
)


Ordering

Fitness



Optimized

Ordering

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Elitist

•
Chromosome representation

–
Integer permutation ordering

–
Sample chromosome in a BBN of 5 nodes might look like: 3 1 2 0 4

•
Seeding

–
Random shuffle

•
Operators

–
Order crossover

–
Swap mutation

•
Fitness

–
RMSE

•
Job farm

–
Java
-
based; Utilize many machines regardless of OS




Properties of the Genetic Algorithm

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Histogram of
estimated

fitness for all 8! = 40320

permutations of
Asia

variables.

•
Not encouraging

–
Bad fitness function

or bad evidence b.v.

–
Many graph errors

GASLEAK results

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
SLAM GA
–

Structure Learning Adjacency Matrix Genetic Algorithm

•
Initial population
-

tried several approaches:

–
Completely Random Bayesian Networks (Box
-
Muller, Max parents)

–
Many illegal structures; wrote fixCycles algorithm.

–
Random networks generated from parents pre
-
selected by the Restrict
phase of Sparse Candidate

–
Performed better than random

–
Aggregate of
k

learned networks from K2 given random orderings (cycles
eliminated)
–

Best approach

Master’s Thesis: SLAM GA

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

For small networks,
k
=1 is best. For larger networks,
k=2

is best.

D

K2

Random Order


K2

Random Order

Aggregator

BBN

BBN

K2

Random Order

BBN

.

.

.

.

Training Data


Aggregate BBN


K2 Manager


BBN

1

2

k

Aggregator Instantiater

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Chromosome representation

–
Edge matrix
–

n^2 bits

–
Each bit represents a parent edge to node.

–
1 = parent, 0 = not parent

•
Operators

–
Crossover: Swap parents, fix cycles.

SLAM GA

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

SLAM GA: Crossover

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

•
Chromosome representation

–
Edge matrix
–

n^2

–
Each bit represents a parent edge to node.

–
1 = parent, 0 = not parent

•
Operators

–
Crossover: Swap parents, fix cycles.

–
Mutation: Reverse, delete, or add a random number of edges. Fix cycles.

•
Fitness

–
Total
Bayesian Dirichlet equivalence

score for each node


SLAM GA

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Asia

Best of first generation

Actual

15 Graph Errors

1 Graph Error

Learned network

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
–

Asia

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Poker

Best of first generation

Actual

11 Graph Errors

2 Graph Errors

Learned network

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Poker

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Golf

Best of first generation

Actual

11 Graph Errors

4 Graph Errors

Learned network

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Golf

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
–

Boerlage92

Initial

Actual

Learned network

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Boerlage92

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Results
-

Alarm

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Final Fitness Values

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

K2 vs. SLAM GA

•
K2:

–
Very good if ordering is known

–
Ordering is often not known

–
Greedy, very dependent on ordering.


•
SLAM GA

–
Stochastic; falls out of local optima trap

–
Can improve on bad structures learned by K2

–
Takes much longer than K2


Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

GASLEAK vs. SLAM GA

•
GASLEAK:

–
Gold network never recovered

–
Much more computationally
-
expensive

–
K2 is run on each [new] individual each generation

–
Each chromosome must be scored

–
Final network has many graph errors

•
SLAM GA

–
For small networks, gold standard network often recovered.

–
Relatively few graph errors for final network.

–
Less computationally intensive

–
Initial population most expensive

–
Each chromosome must be scored

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

SLAM GA: Ramifications

•
Effective structure learning algorithm

–
Ideal for small networks

•
Improvement over GASLEAK

–
SLAM GA faster in spite of same GA parameters

–
SLAM GA more accurate

•
Improvement over K2

•
Aggregate algorithm produces better initial population

•
Parent
-
swapping crossover technique effective

–
Diversifies search space while retaining past information


Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

SLAM GA: Future Work

•
Parameter tweaking

•
Better fitness function

–
Several ‘bad’ structures score better than gold standard

–
GA works fine

•
‘Intelligent’ mutation operator

–
Add edges from pre
-
qualified set of candidate parents

•
New instantiation methods

–
Use GASLEAK

–
Other structure
-
learning algorithms

•
Scalability

–
Job farm

Kansas State University

Department of Computing and Information Sciences

Ben Perry
–

M.S. thesis defense

Summary

•
Bayesian Network

•
Genetic Algorithms

•
Learning Structure: K2, Sparse Candidate

•
GASLEAK

•
SLAM GA