Interestingness

of

Frequent

Itemsets

Using

Bayesian

Networks

as

Background

Knowledge

The

paper

presents

a

method

for

pruning

frequent

itemsets

based

on

background

knowledge

represented

by

a

Bayesian

network

The

interestingness

of

an

itemset

is

defined

as

the

absolute

difference

between

its

support

estimated

from

data

and

from

the

Bayesian

network

Efficient

algorithms

are

presented

for

finding

interestingness

of

a

collection

of

frequent

itemsets

and

for

finding

all

attribute

sets

with

a

given

minimum

interestingness

Practical

usefulness

of

the

algorithms

and

their

efficiency

have

been

verified

experimentally

H.

-LSB-

Database

Management

-RSB-

Database

Applications-data

mining

Algorithms

Finding

frequent

itemsets

and

association

rules

in

database

tables

has

been

an

active

research

area

in

recent

years

Unfortunately

the

practical

usefulness

of

the

approach

is

limited

by

huge

number

of

patterns

usually

discovered

For

larger

databases

many

thousands

of

association

rules

may

be

produced

when

minimum

support

is

low

This

creates

a

secondary

data

mining

problem

after

mining

the

data

we

are

now

compelled

to

mine

the

discovered

patterns

The

problem

has

been

addressed

in

literature

mainly

in

the

context

of

association

rules

where

the

two

main

approaches

are

Permission

to

make

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or

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copies

of

all

or

part

of

this

work

for

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or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page

To

copy

otherwise

to

republish

to

post

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servers

or

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redistribute

to

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requires

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specific

permission

and/or

a

fee

KDD

August

Seattle

Washington

USA

Copyright

ACM

sorting

rules

based

on

some

interestingness

measure

and

pruning

aiming

at

removing

redundant

rules

Full

review

of

such

methods

is

beyond

the

scope

of

this

paper

Overviews

of

interestingness

measures

can

be

found

for

example

in

-LSB-

-RSB-

some

of

the

papers

on

rule

pruning

are

-LSB-

-RSB-

Many

interestingness

measures

are

based

on

the

divergence

between

true

probability

distributions

and

distributions

obtained

under

the

independence

assumption

Pruning

methods

are

usually

based

on

comparing

the

confidence

of

a

rule

to

the

confidence

of

rules

related

to

it

The

main

drawback

of

those

methods

is

that

they

tend

to

generate

rules

that

are

either

obvious

or

have

already

been

known

by

the

user

This

is

to

be

expected

since

the

most

striking

patterns

which

those

methods

select

can

also

easily

be

discovered

using

traditional

methods

or

are

known

directly

from

experience

We

believe

that

the

proper

way

to

address

the

problem

is

to

include

users

background

knowledge

in

the

process

The

patterns

which

diverge

the

most

from

that

background

knowledge

are

deemed

most

interesting

Discovered

patterns

can

later

be

applied

to

improve

the

background

knowledge

itself

Many

approaches

to

using

background

knowledge

in

machine

learning

are

focused

on

using

background

knowledge

to

speed

up

the

hypothesis

discovery

process

and

not

on

discovering

interesting

patterns

Those

methods

often

assume

strict

logical

relationships

not

probabilistic

ones

Examples

are

knowledge

based

neural

networks

-LRB-

KBANNs

-RRB-

and

uses

of

background

knowledge

in

Inductive

Logic

Programming

See

Chapter

in

-LSB-

-RSB-

for

an

overview

of

those

methods

and

a

list

of

further

references

Tuzhilin

et

al.

-LSB-

-RSB-

worked

on

applying

background

knowledge

to

finding

interesting

rules

In

-LSB-

-RSB-

interestingness

measures

are

presented

which

take

into

account

prior

beliefs

in

another

paper

-LSB-

-RSB-

the

authors

present

an

algorithm

for

selecting

a

minimum

set

of

interesting

rules

given

background

knowledge

The

methods

used

in

those

papers

are

local

that

is

they

do

n't

use

a

full

joint

probability

of

the

data

Instead

interestingness

of

a

rule

is

evaluated

using

rules

in

the

background

knowledge

with

the

same

consequent

If

no

such

knowledge

is

present

for

a

given

rule

the

rule

is

considered

uninteresting

This

makes

it

impossible

to

take

into

account

transitivity

Indeed

in

the

presence

of

the

background

knowledge

represented

by

the

rules

A

B

and

Research

Track

Paper

B

C

the

rule

A

C

is

uninteresting

However

this

can

not

be

discovered

locally

See

-LSB-

-RSB-

for

a

detailed

discussion

of

advantages

of

global

versus

local

methods

Some

more

comparisons

can

be

found

in

-LSB-

-RSB-

In

this

paper

we

present

a

method

of

finding

interesting

patterns

using

background

knowledge

represented

by

a

Bayesian

network

The

main

advantage

of

Bayesian

networks

is

that

they

concisely

represent

full

joint

probability

distributions

and

allow

for

practically

feasible

probabilistic

inference

from

those

distributions

-LSB-

-RSB-

Other

advantages

include

the

ability

to

represent

causal

relationships

easy

to

understand

graphical

structure

as

well

as

wide

availability

of

modelling

tools

Bayesian

networks

are

also

easy

to

modify

by

adding

or

deleting

edges

We

opt

to

compute

interestingness

of

frequent

itemsets

instead

of

association

rules

agreeing

with

-LSB-

-RSB-

that

directions

of

dependence

should

be

decided

by

the

user

based

on

her

experience

and

not

suggested

by

interestingness

measures

Our

approach

works

by

estimating

supports

of

itemsets

from

Bayesian

networks

and

comparing

thus

estimated

supports

with

the

data

Itemsets

with

strongly

diverging

supports

are

considered

interesting

Further

definitions

of

interestingness

exploiting

Bayesian

network

's

structure

are

presented

as

well

as

efficient

methods

for

computing

interestingness

of

large

numbers

of

itemsets

and

for

finding

all

attribute

sets

with

given

minimum

interestingness

There

are

some

analogies

between

mining

emerging

patterns

-LSB-

-RSB-

and

our

approach

the

main

differences

being

that

in

our

case

a

Bayesian

network

is

used

instead

of

a

second

dataset

and

that

we

use

a

different

measure

for

comparing

supports

Due

to

those

differences

our

problem

requires

a

different

approach

and

a

different

set

of

algorithms

Database

attributes

will

be

denoted

with

uppercase

letters

A

B

C

domain

of

an

attribute

A

will

be

denoted

by

Dom

-LRB-

A

-RRB-

In

this

paper

we

are

only

concerned

with

categorical

attributes

that

is

attributes

with

finite

domains

Sets

of

attributes

will

be

denoted

with

uppercase

letters

I

J

We

often

use

database

notation

for

representing

sets

of

attributes

i.e.

I

A

A

A

k

instead

of

the

set

theoretical

notation

-LCB-

A

A

A

k

-RCB-

Domain

of

an

attribute

set

I

A

A

A

k

is

defined

as

Dom

-LRB-

I

-RRB-

Dom

-LRB-

A

-RRB-

Dom

-LRB-

A

-RRB-

Dom

-LRB-

A

k

-RRB-

Values

from

domains

of

attributes

and

attribute

sets

are

denoted

with

corresponding

lowercase

boldface

letters

e.g.

i

Dom

-LRB-

I

-RRB-

Let

P

I

denote

a

joint

probability

distribution

of

the

attribute

set

I.

Similarly

let

P

I

J

be

a

distribution

of

I

conditioned

on

J

When

used

in

arithmetic

operations

such

distributions

will

be

treated

as

functions

of

attributes

in

I

and

I

J

respectively

with

values

in

the

interval

-LSB-

-RSB-

For

example

P

I

-LRB-

i

-RRB-

denotes

the

probability

that

I

i.

Let

P

I

be

a

probability

distribution

and

let

J

I.

Denote

by

P

J

I

the

marginalization

of

P

I

onto

J

that

is

P

J

I

X

I

J

P

I

-LRB-

-RRB-

where

the

summation

is

over

the

domains

of

all

variables

from

I

J.

Probability

distributions

estimated

from

data

will

be

denoted

by

adding

a

hat

symbol

e.g.

P

I

An

itemset

is

a

pair

-LRB-

I

i

-RRB-

where

I

is

an

attribute

set

and

i

Dom

-LRB-

I

-RRB-

The

support

of

an

itemset

-LRB-

I

i

-RRB-

is

defined

as

supp

-LRB-

I

i

-RRB-

P

I

-LRB-

i

-RRB-

where

the

probability

is

estimated

from

some

dataset

An

itemset

is

called

frequent

if

its

support

is

greater

than

or

equal

to

some

user

defined

threshold

minsupp

Finding

all

frequent

itemsets

in

a

given

database

table

is

a

well

known

datamining

problem

-LSB-

-RSB-

A

Bayesian

network

BN

over

a

set

of

attributes

H

A

A

n

is

a

directed

acyclic

graph

BN

-LRB-

V

E

-RRB-

with

the

set

of

vertices

V

-LCB-

V

A

V

A

n

-RCB-

corresponding

to

attributes

of

H

and

a

set

of

edges

E

V

V

where

each

vertex

V

A

i

has

associated

a

conditional

probability

distribution

P

A

i

par

i

where

par

i

-LCB-

A

j

-LRB-

V

A

j

V

A

i

-RRB-

E

-RCB-

is

the

set

of

attributes

corresponding

to

parents

of

V

A

i

in

G.

See

-LSB-

-RSB-

for

a

detailed

discussion

of

Bayesian

networks

A

Bayesian

network

BN

over

H

uniquely

defines

a

joint

probability

distribution

P

BN

H

n

Y

i

P

A

i

par

i

of

H.

For

I

H

the

distribution

over

I

marginalized

from

P

BN

H

will

be

denoted

by

P

BN

I

P

BN

I

P

BN

H

I

Let

us

first

define

the

support

of

an

itemset

-LRB-

I

i

-RRB-

in

a

Bayesian

network

BN

as

supp

BN

-LRB-

I

i

-RRB-

P

BN

I

-LRB-

i

-RRB-

Let

BN

be

a

Bayesian

network

over

an

attribute

set

H

and

let

-LRB-

I

i

-RRB-

be

an

itemset

such

that

I

H

The

interestingness

of

the

itemset

-LRB-

I

i

-RRB-

with

respect

to

BN

is

defined

as

I

BN

-LRB-

I

i

-RRB-

supp

-LRB-

I

i

-RRB-

supp

BN

-LRB-

I

i

-RRB-

that

is

the

absolute

difference

between

the

support

of

the

itemset

estimated

from

data

and

the

estimate

of

this

support

made

from

the

Bayesian

network

BN

In

the

remaining

part

of

the

paper

we

assume

that

interestingness

is

always

computed

with

respect

to

a

Bayesian

network

BN

and

the

subscript

is

omitted

An

itemset

is

interesting

if

its

interestingness

is

greater

than

or

equal

to

some

user

specified

threshold

A

frequent

interesting

itemset

represents

a

frequently

occurring

-LRB-

due

to

minimum

support

requirement

-RRB-

pattern

in

the

database

whose

probability

is

significantly

different

from

what

it

is

believed

to

be

based

on

the

Bayesian

network

model

An

alternative

would

be

to

use

supp

-LRB-

I

i

-RRB-

supp

BN

-LRB-

I

i

-RRB-

as

the

measure

of

interestingness

-LSB-

-RSB-

We

decided

to

use

absolute

difference

instead

of

a

quotient

since

we

found

it

to

be

more

robust

especially

when

both

supports

are

small

One

could

think

of

applying

our

approach

to

association

rules

with

the

difference

in

confidences

as

a

measure

of

interestingness

but

as

mentioned

in

the

Introduction

we

think

Research

Track

Paper

that

patterns

which

do

not

suggest

a

direction

of

influence

are

more

appropriate

Since

in

Bayesian

networks

dependencies

are

modelled

using

attributes

not

itemsets

it

will

often

be

easier

to

talk

about

interesting

attribute

sets

especially

when

the

discovered

interesting

patterns

are

to

be

used

to

update

the

background

knowledge

Definition

Let

I

be

an

attribute

set

The

interestingness

of

I

is

defined

as

I

-LRB-

I

-RRB-

max

iDom

-LRB-

I

-RRB-

I

-LRB-

I

i

-RRB-

-LRB-

-RRB-

analogously

I

is

interesting

if

I

-LRB-

I

-RRB-

An

alternative

approach

would

be

to

use

generalizations

of

Bayesian

networks

allowing

dependencies

to

vary

for

different

values

of

attributes

see

-LSB-

-RSB-

and

deal

with

itemset

interestingness

directly

Extensions

to

the

Definition

of

Interestingness

Even

though

applying

the

above

definition

and

sorting

attribute

sets

on

their

interestingness

works

well

in

practice

there

might

still

be

a

large

number

of

patterns

retained

especially

if

the

background

knowledge

is

not

well

developed

and

large

number

of

attribute

sets

have

high

interestingness

values

This

motivates

the

following

two

definitions

Definition

An

attribute

set

I

is

hierarchically

interesting

if

it

is

interesting

and

of

its

proper

subsets

is

interesting

The

idea

is

to

prevent

large

attribute

sets

from

becoming

interesting

when

the

true

cause

of

them

being

interesting

lies

in

their

subsets

There

is

also

another

problem

with

Definition

Consider

a

Bayesian

network

A

B

where

nodes

A

and

B

have

respective

probability

distributions

P

A

and

P

B

A

attached

Suppose

also

that

A

is

interesting

In

this

case

even

if

P

B

A

is

the

same

as

P

B

A

attribute

sets

B

and

AB

may

be

considered

interesting

Below

we

present

a

definition

of

interestingness

aiming

at

preventing

such

situations

A

vertex

V

is

an

ancestor

of

a

vertex

W

in

a

directed

graph

G

if

there

is

a

directed

path

from

V

to

W

in

G

The

set

of

ancestors

of

a

vertex

V

in

a

graph

G

is

denoted

by

anc

-LRB-

V

-RRB-

Moreover

let

us

denote

by

anc

-LRB-

I

-RRB-

the

set

of

all

ancestor

attributes

in

BN

of

an

attribute

set

I

More

formally

anc

-LRB-

I

-RRB-

-LCB-

A

i

I

V

A

i

anc

-LRB-

V

A

j

-RRB-

in

BN

for

some

A

j

I

-RCB-

Definition

An

attribute

set

I

is

topologically

interesting

if

it

is

interesting

and

there

is

no

attribute

set

J

such

that

J

anc

-LRB-

I

-RRB-

I

and

I

J

and

J

is

interesting

The

intention

here

is

to

prevent

interesting

attribute

sets

from

causing

all

their

successors

in

the

Bayesian

network

-LRB-

and

the

supersets

of

their

successors

-RRB-

to

become

interesting

in

a

cascading

fashion

To

see

why

condition

is

necessary

consider

a

Bayesian

network

A

X

B

Suppose

that

there

is

a

dependency

between

A

and

B

in

data

which

makes

AB

interesting

Now

however

ABX

may

also

become

interesting

-LRB-

even

if

P

A

X

and

P

B

X

are

correct

in

the

network

-RRB-

and

cause

AB

to

be

pruned

Condition

prevents

AB

from

being

pruned

and

ABX

from

becoming

interesting

Notice

that

topological

interestingness

is

stricter

than

hierarchical

interestingness

Indeed

if

J

I

is

interesting

then

it

satisfies

all

the

above

conditions

and

makes

I

not

topologically

interesting

In

this

section

we

present

algorithms

using

the

definition

of

interestingness

introduced

in

the

previous

section

to

select

interesting

itemsets

or

attribute

sets

We

begin

by

describing

a

procedure

for

computing

marginal

distributions

for

a

large

collection

of

attribute

sets

from

a

Bayesian

network

Computing

a

Large

Number

of

Marginal

Distributions

from

a

Bayesian

Network

Computing

the

interestingness

of

a

large

number

of

frequent

itemsets

requires

the

computation

of

a

large

number

of

marginal

distributions

from

a

Bayesian

network

The

problem

has

been

addressed

in

literature

mainly

in

the

context

of

finding

marginals

for

every

attribute

-LSB-

-RSB-

while

here

we

have

to

find

marginals

for

multiple

overlapping

sets

of

attributes

The

approach

taken

in

this

paper

is

outlined

below

The

problem

of

computing

marginal

distributions

from

a

Bayesian

network

is

known

to

be

NP-hard

nevertheless

in

most

cases

the

network

structure

can

be

exploited

to

speed

up

the

computations

Here

we

use

exact

methods

for

computing

the

marginals

Approximate

methods

like

Gibbs

sampling

are

an

interesting

topic

for

future

work

Best

known

approaches

to

exact

marginalizations

are

join

trees

-LSB-

-RSB-

and

bucket

elimination

-LSB-

-RSB-

We

chose

bucket

elimination

method

which

is

easier

to

implement

and

according

to

-LSB-

-RSB-

as

efficient

as

join

tree

based

methods

Also

join

trees

are

mainly

useful

for

computing

marginals

for

single

attributes

and

not

for

sets

of

attributes

The

bucket

elimination

method

which

is

based

on

the

distributive

law

proceeds

by

first

choosing

a

variable

ordering

and

then

applying

distributive

law

repeatedly

to

simplify

the

summation

For

example

suppose

that

a

joint

distribution

of

a

Bayesian

network

over

H

ABC

is

expressed

as

P

BN

ABC

P

A

P

B

A

P

C

A

and

we

want

to

find

P

BN

A

We

need

to

compute

the

sum

X

B

X

C

P

A

P

B

A

P

C

A

Research

Track

Paper

which

can

be

rewritten

as

P

A

X

b

Dom

-LRB-

B

-RRB-

P

B

A

A

X

c

Dom

-LRB-

C

-RRB-

P

C

A

A.

Assuming

that

domains

of

all

attributes

have

size

computing

the

first

sum

directly

requires

additions

and

multiplications

while

the

second

sum

requires

only

additions

and

multiplications

The

expression

is

interpreted

as

a

tree

of

buckets

each

bucket

is

either

a

single

probability

distribution

or

a

sum

over

a

single

attribute

taken

over

a

product

of

its

child

buckets

in

the

tree

In

the

example

above

a

special

root

bucket

without

summation

could

be

introduced

for

completeness

In

most

cases

the

method

significantly

reduces

the

time

complexity

of

the

problem

An

important

problem

is

choosing

the

right

variable

ordering

Unfortunately

that

problem

is

itself

NP-hard

We

thus

adopt

a

heuristic

which

orders

variables

according

to

the

decreasing

number

of

factors

in

the

product

depending

on

each

variable

A

detailed

discussion

of

the

method

can

be

found

in

-LSB-

-RSB-

Although

bucket

elimination

can

be

used

to

obtain

supports

of

itemsets

directly

-LRB-

i.e.

P

I

-LRB-

i

-RRB-

-RRB-

we

use

it

to

obtain

complete

marginal

distributions

This

way

we

can

directly

apply

marginalization

to

obtain

distributions

for

subsets

of

I

-LRB-

see

below

-RRB-

Since

bucket

elimination

is

performed

repeatedly

we

use

memoization

to

speed

it

up

as

suggested

in

-LSB-

-RSB-

We

remember

each

partial

sum

and

reuse

it

if

possible

In

the

example

above

P

b

Dom

-LRB-

B

-RRB-

P

B

A

P

c

Dom

-LRB-

C

-RRB-

P

C

A

and

the

computed

P

BN

A

would

have

been

remembered

Another

method

of

obtaining

a

marginal

distribution

P

I

is

marginalizing

it

from

P

J

where

I

J

using

Equation

-LRB-

-RRB-

provided

that

P

J

is

already

known

If

J

I

is

small

this

procedure

is

almost

always

more

efficient

than

bucket

elimination

so

whenever

some

P

I

is

computed

by

bucket

elimination

distributions

of

all

subsets

of

I

are

computed

using

Equation

-LRB-

-RRB-

Definition

Let

C

be

a

collection

of

attribute

sets

The

positive

border

of

C

-LSB-

-RSB-

denoted

by

Bd

-LRB-

C

-RRB-

is

the

collection

of

those

sets

from

C

which

have

no

proper

superset

in

C

Bd

-LRB-

C

-RRB-

-LCB-

I

C

there

is

no

J

C

such

that

I

J

-RCB-

It

is

clear

from

the

discussion

above

that

we

only

need

to

use

bucket

elimination

to

compute

distributions

of

itemsets

in

the

positive

border

We

are

going

to

go

further

than

this

we

will

use

bucket

elimination

to

obtain

supersets

of

sets

in

the

positive

border

and

then

use

Equation

-LRB-

-RRB-

to

obtain

marginals

even

for

sets

in

the

positive

border

Experiments

show

that

this

approach

can

give

substantial

savings

especially

when

many

overlapping

attribute

sets

from

the

positive

border

can

be

covered

by

a

single

set

only

slightly

larger

then

the

covered

ones

The

algorithm

for

selecting

the

marginal

distribution

to

compute

is

motivated

by

the

algorithm

from

-LSB-

-RSB-

for

computing

views

that

should

be

materialized

for

OLAP

query

processing

Bucket

elimination

corresponds

to

creating

a

materialized

view

and

marginalizing

thus

obtained

distribution

to

answering

OLAP

queries

We

first

need

to

define

costs

of

marginalization

and

bucket

elimination

In

our

case

the

cost

is

defined

as

the

total

number

of

additions

and

multiplications

used

to

compute

the

marginal

distribution

The

cost

of

marginalizing

P

J

from

P

I

J

I

using

Equation

-LRB-

-RRB-

is

cost

-LRB-

P

J

I

-RRB-

Dom

-LRB-

J

-RRB-

-LRB-

Dom

-LRB-

I

J

-RRB-

-RRB-

It

follows

from

the

fact

that

each

value

of

P

J

I

requires

adding

Dom

-LRB-

I

J

-RRB-

values

from

P

I

The

cost

of

bucket

elimination

can

be

computed

cheaply

without

actually

executing

the

procedure

Each

bucket

is

either

an

explicitly

given

probability

distribution

or

computes

a

sum

over

a

single

variable

of

a

product

of

functions

-LRB-

computed

in

buckets

contained

in

it

-RRB-

explicitly

represented

as

multidimensional

tables

see

-LSB-

-RSB-

for

details

If

the

bucket

is

an

explicitly

given

probability

distribution

the

cost

is

of

course

Consider

now

a

bucket

b

containing

child

buckets

b

b

n

yielding

functions

f

f

n

respectively

Let

Var

-LRB-

f

i

-RRB-

the

set

of

attributes

on

which

f

i

depends

Let

f

f

f

f

n

denote

the

product

of

all

factors

in

b

We

have

Var

-LRB-

f

-RRB-

n

i

Var

-LRB-

f

i

-RRB-

and

since

each

value

of

f

requires

n

multiplications

computing

f

requires

Dom

-LRB-

Var

-LRB-

f

-RRB-

-RRB-

-LRB-

n

-RRB-

multiplications

Let

A

b

be

the

attribute

over

which

summation

in

b

takes

place

Computing

the

sum

will

require

Dom

-LRB-

Var

-LRB-

f

-RRB-

-LCB-

A

b

-RCB-

-RRB-

-LRB-

Dom

-LRB-

A

b

-RRB-

-RRB-

additions

So

the

total

cost

of

computing

the

function

in

bucket

b

-LRB-

including

costs

of

computing

its

children

-RRB-

is

thus

cost

-LRB-

b

-RRB-

n

X

i

cost

-LRB-

b

i

-RRB-

Dom

-LRB-

Var

-LRB-

f

-RRB-

-RRB-

-LRB-

n

-RRB-

Dom

-LRB-

Var

-LRB-

f

-RRB-

-LCB-

A

b

-RCB-

-RRB-

-LRB-

Dom

-LRB-

A

b

-RRB-

-RRB-

The

cost

of

computing

P

BN

I

through

bucket

elimination

denoted

cost

BE

-LRB-

P

BN

I

-RRB-

is

the

cost

of

the

root

bucket

of

the

summation

used

to

compute

P

BN

I

Let

C

be

a

collection

of

attribute

sets

The

gain

of

using

bucket

elimination

to

find

P

BN

I

for

some

I

while

computing

interestingness

of

attribute

sets

from

C

can

be

expressed

as

gain

-LRB-

I

-RRB-

cost

BE

-LRB-

P

BN

I

-RRB-

X

J

Bd

-LRB-

C

-RRB-

J

I

hcost

BE

-LRB-

P

BN

J

-RRB-

cost

-LRB-

P

BN

I

J

-RRB-

i

An

attribute

set

to

which

bucket

elimination

will

be

applied

is

found

using

a

greedy

procedure

by

adding

in

each

itera-tion

the

attribute

giving

the

highest

increase

of

gain

The

complete

algorithm

is

presented

in

Figure

Computing

The

Interestingness

of

a

Collection

of

Itemsets

First

we

present

an

algorithm

for

computing

interestingness

of

all

itemsets

in

a

given

collection

Its

a

simple

application

of

the

algorithm

in

Figure

It

is

useful

if

we

already

Research

Track

Paper

Input

collection

of

attribute

sets

C

Bayesian

network

BN

Output

distributions

P

BN

I

for

all

I

C

S

Bd

-LRB-

C

-RRB-

while

S

I

an

attribute

set

from

S.

for

A

in

H

I

compute

gain

-LRB-

I

-LCB-

A

-RCB-

-RRB-

pick

A

for

which

the

gain

in

Step

was

maximal

if

gain

-LRB-

I

-LCB-

A

-RCB-

-RRB-

gain

-LRB-

I

-RRB-

I

I

-LCB-

A

-RCB-

goto

compute

P

BN

I

from

BN

using

bucket

elimination

compute

P

BN

I

J

for

all

J

S

J

I

using

Equation

-LRB-

-RRB-

remove

from

S

all

attribute

sets

included

in

I

compute

P

BN

J

for

all

J

C

Bd

-LRB-

C

-RRB-

using

Equation

-LRB-

-RRB-

Figure

Algorithm

for

computing

a

large

number

of

marginal

distributions

from

a

Bayesian

network

have

a

collection

of

itemsets

-LRB-

e.g.

all

frequent

itemsets

found

in

a

database

table

-RRB-

and

want

to

select

those

which

are

the

most

interesting

The

algorithm

is

given

below

Input

collection

of

itemsets

K

supports

of

all

itemsets

in

K

Bayesian

network

BN

Output

interestingness

of

all

itemsets

in

K.

C

-LCB-

I

-LRB-

I

i

-RRB-

K

for

some

i

Dom

-LRB-

I

-RRB-

-RCB-

compute

P

BN

I

for

all

I

C

using

algorithm

in

Figure

compute

interestingness

of

all

itemsets

in

K

using

distributions

computed

in

step

Finding

All

Attribute

Sets

With

Given

Minimum

Interestingness

In

this

section

we

will

present

an

algorithm

for

finding

all

attribute

sets

with

interestingness

greater

than

or

equal

to

a

specified

threshold

given

a

dataset

and

a

Bayesian

network

BN

Let

us

first

make

an

observation

Observation

If

an

itemset

-LRB-

I

i

-RRB-

has

interestingness

greater

than

or

equal

to

with

respect

to

a

Bayesian

network

BN

then

its

support

must

be

greater

than

or

equal

to

in

either

the

data

or

in

BN

Moreover

if

an

attribute

set

is

interesting

by

definition

at

least

one

of

its

itemsets

must

be

interesting

It

follows

that

if

an

attribute

set

is

interesting

then

one

of

its

itemsets

must

be

frequent

with

minimum

support

either

in

the

data

or

in

the

Bayesian

network

Input

Bayesian

network

BN

minimum

support

minsupp

Output

itemsets

whose

support

in

BN

is

minsupp

k

Cand

-LCB-

-LRB-

I

i

-RRB-

I

-RCB-

compute

supp

BN

-LRB-

I

i

-RRB-

for

all

-LRB-

I

i

-RRB-

Cand

using

the

algorithm

in

Figure

F

req

k

-LCB-

-LRB-

I

i

-RRB-

Cand

supp

BN

-LRB-

I

i

-RRB-

minsupp

-RCB-

Cand

generate

new

candidates

from

F

req

k

remove

itemsets

with

infrequent

subsets

from

Cand

k

k

goto

Figure

The

AprioriBN

algorithm

The

algorithm

works

in

two

stages

First

all

frequent

itemsets

with

minimum

support

are

found

in

the

dataset

and

their

interestingness

is

computed

The

first

stage

might

have

missed

itemsets

which

are

interesting

but

do

n't

have

sufficient

support

in

the

data

In

the

second

stage

all

itemsets

frequent

in

the

Bayesian

network

are

found

and

their

supports

in

the

data

are

computed

using

an

extra

database

scan

To

find

all

itemsets

frequent

in

the

Bayesian

network

we

use

the

Apriori

algorithm

-LSB-

-RSB-

with

a

modified

support

counting

part

which

we

call

AprioriBN

The

sketch

of

the

algorithm

is

shown

in

Figure

except

for

step

it

is

identical

to

the

original

algorithm

We

now

have

all

the

elements

needed

to

present

the

algorithm

for

finding

all

interesting

attribute

sets

which

is

given

in

Figure

Step

of

the

algorithm

can

reuse

marginal

distributions

found

in

step

to

speed

up

the

computations

Notice

that

it

is

always

possible

to

compute

interestingness

of

every

itemset

in

step

since

both

supports

of

each

itemset

will

be

computed

either

in

steps

and

or

in

steps

and

The

authors

implemented

hierarchical

and

topological

interestingness

as

a

postprocessing

step

They

could

however

be

used

to

prune

the

attribute

sets

which

are

not

interesting

without

evaluating

their

distributions

thus

providing

a

potentially

large

speedup

in

the

computations

We

plan

to

investigate

that

in

the

future

In

this

section

we

present

experimental

evaluation

of

the

method

One

problem

we

were

faced

with

was

the

lack

of

publicly

available

datasets

with

nontrivial

background

knowledge

that

could

be

represented

as

a

Bayesian

network

The

UCI

Machine

Learning

repository

contains

a

few

datasets

with

background

knowledge

-LRB-

Japanese

credit

molecular

biology

-RRB-

but

they

are

aimed

primarily

at

Inductive

Logic

Programming

the

relationships

are

logical

rather

than

probabilistic

only

relationships

involving

the

class

attribute

are

included

These

examples

are

of

little

value

for

our

approach

We

have

thus

used

networks

constructed

using

our

own

Research

Track

Paper

Input

Bayesian

network

BN

dataset

interestingness

threshold

Output

all

attribute

sets

with

interestingness

at

least

and

some

of

the

attribute

sets

with

lower

interestingness

K

-LCB-

-LRB-

I

i

-RRB-

supp

-LRB-

I

i

-RRB-

-RCB-

-LRB-

using

Apriori

algorithm

-RRB-

C

-LCB-

I

-LRB-

I

i

-RRB-

K

for

some

i

Dom

-LRB-

I

-RRB-

-RCB-

compute

P

BN

I

for

all

I

C

using

algorithm

in

Figure

K

-LCB-

-LRB-

I

i

-RRB-

supp

BN

-LRB-

I

i

-RRB-

-RCB-

-LRB-

using

AprioriBN

algorithm

-RRB-

compute

support

in

data

for

all

itemsets

in

K

K

by

scanning

the

dataset

compute

interestingness

of

all

itemsets

in

K

K

C

-LCB-

I

-LRB-

I

i

-RRB-

K

for

some

i

Dom

-LRB-

I

-RRB-

-RCB-

compute

interestingness

of

all

attribute

sets

I

in

C

C

I

-LRB-

I

-RRB-

max

-LCB-

I

-LRB-

I

i

-RRB-

-LRB-

I

i

-RRB-

K

K

i

Dom

-LRB-

I

-RRB-

-RCB-

Figure

Algorithm

for

finding

all

interesting

attribute

sets

common-sense

knowledge

as

well

as

networks

learned

from

data

An

Illustrative

Example

We

first

present

a

simple

example

demonstrating

the

usefulness

of

the

method

We

use

the

KSL

dataset

of

Danish

year

olds

distributed

with

the

DEAL

Bayesian

network

package

-LSB-

-RSB-

There

are

nine

attributes

described

in

Table

related

to

the

person

's

general

health

and

lifestyle

All

continuous

attributes

have

been

discretized

into

levels

using

the

equal

weight

method

FEV

Forced

ejection

volume

of

person

's

lungs

Kol

Cholesterol

Hyp

Hypertension

-LRB-

no/yes

-RRB-

BMI

Body

Mass

Index

Smok

Smoking

-LRB-

no/yes

-RRB-

Alc

Alcohol

consumption

-LRB-

seldom/frequently

-RRB-

Work

Working

-LRB-

yes/no

-RRB-

Sex

male/female

Year

Survey

year

-LRB-

-RRB-

Table

Attributes

of

the

KSL

dataset

We

began

by

designing

a

network

structure

based

on

au-thors

-LRB-

non-expert

-RRB-

knowledge

The

network

structure

is

given

in

Figure

Since

we

were

not

sure

about

the

relation

of

cholesterol

to

other

attributes

we

left

it

unconnected

Conditional

probabilities

were

estimated

directly

from

the

KSL

dataset

Note

that

this

is

a

valid

approach

since

even

when

the

conditional

probabilities

match

the

data

perfectly

interesting

patterns

can

still

be

found

because

the

network

structure

usually

is

not

capable

of

representing

the

full

joint

distribution

of

the

data

The

interesting

patterns

can

then

be

used

to

update

the

network

's

structure

Of

course

if

both

the

structure

and

the

conditional

probabilities

are

given

by

a

-RRB-

b

-RRB-

Figure

Network

structures

for

the

KSL

dataset

constructed

by

the

authors

the

expert

then

the

discovered

patterns

can

be

used

to

update

both

the

network

's

structure

and

conditional

probabilities

We

applied

the

algorithm

for

finding

all

interesting

attribute

sets

to

the

KSL

dataset

and

the

network

using

the

threshold

of

The

attribute

sets

returned

were

sorted

by

interestingness

and

top

results

were

kept

The

two

most

interesting

attribute

sets

were

-LCB-

F

EV

Sex

-RCB-

with

interestingness

and

-LCB-

Alc

Y

ear

-RCB-

with

interestingness

Indeed

it

is

known

-LRB-

see

-LSB-

-RSB-

-RRB-

that

women

's

lungs

are

on

average

smaller

than

men

's

lungs

so

sex

influences

the

forced

ejection

volume

-LRB-

FEV

-RRB-

much

more

than

smoking

does

-LRB-

which

we

thought

was

the

primary

influence

-RRB-

This

fact

although

not

new

in

general

was

overlooked

by

the

authors

and

we

suspect

that

due

to

large

amount

of

literature

on

harmful

effects

of

smoking

it

might

have

been

overlooked

by

many

domain

experts

This

proves

the

high

value

of

our

approach

for

verification

of

Bayesian

network

models

The

data

itself

implied

a

growth

in

alcohol

consumption

between

and

which

we

considered

to

be

a

plausible

finding

We

then

decided

to

modify

the

network

structure

based

on

our

findings

by

adding

edges

Sex

F

EV

and

Y

ear

Alc

One

could

of

course

consider

other

methods

of

modifying

network

structure

like

deleting

edges

or

reversing

their

direction

A

brief

overview

of

more

advanced

methods

of

Bayesian

network

modification

can

be

found

in

-LSB-

Chap

Sect.

-RSB-

Instead

of

adapting

the

network

structure

one

could

keep

the

structure

unchanged

and

tune

conditional

probabilities

in

the

network

instead

see

-LSB-

Chap

Sect.

-RSB-

for

details

Research

Track

Paper

As

a

method

of

scoring

network

structures

we

used

the

natural

logarithm

of

the

probability

of

the

structure

conditioned

on

the

data

see

-LSB-

-RSB-

for

details

on

computing

the

score

The

modified

network

structure

had

the

score

of

which

is

better

than

that

of

the

original

network

With

the

modified

structure

the

most

interesting

attribute

set

was

-LCB-

Kol

Sex

Y

ear

-RCB-

with

interestingness

We

found

in

the

data

that

cholesterol

levels

decreased

between

the

two

years

in

which

the

study

was

made

and

that

cholesterol

level

depends

on

sex

We

found

similar

trends

in

the

U.S.

population

based

on

data

from

American

Heart

Association

-LSB-

-RSB-

Adding

edges

Y

ear

Kol

and

Sex

Kol

improved

the

network

score

to

-LCB-

F

EV

Alc

Y

ear

-RCB-

became

the

most

interesting

attribute

set

with

the

interestingness

of

Its

interestingness

is

however

much

lower

than

that

of

previous

most

interesting

attribute

sets

Also

we

were

not

able

to

get

any

improvement

in

network

score

after

adding

edges

related

to

that

attribute

set

Since

we

were

unable

to

obtain

a

better

network

in

this

case

we

used

topological

pruning

expecting

that

some

other

attribute

sets

might

be

the

true

cause

of

the

observed

discrepancies

Only

four

attribute

sets

given

below

were

topologically

-LCB-

Kol

BM

I

-RCB-

-LCB-

Kol

Alc

-RCB-

-LCB-

Smok

Sex

Y

ear

-RCB-

-LCB-

Alc

W

ork

-RCB-

We

found

all

those

patters

intuitively

valid

but

were

unable

to

obtain

an

improvement

in

the

network

's

score

by

adding

related

edges

Moreover

the

interestingness

values

were

quite

small

We

thus

finished

the

interactive

network

structure

improvement

process

with

the

final

result

given

in

Figure

The

algorithm

was

implemented

in

Python

and

used

on

a

GHz

Pentium

machine

The

computation

of

interestingness

for

this

example

took

only

a

few

seconds

so

an

interactive

use

of

the

program

was

possible

Further

performance

evaluation

is

given

below

Performance

Evaluation

We

now

present

the

performance

evaluation

of

the

algorithm

for

finding

all

attribute

sets

with

given

minimum

interestingness

We

used

the

UCI

datasets

and

Bayesian

networks

learned

from

data

using

B-Course

-LSB-

-RSB-

The

results

are

given

in

Table

The

max

size

column

gives

the

maximum

size

of

frequent

attribute

sets

considered

The

#marginals

column

gives

the

total

number

of

marginal

distributions

computed

from

the

Bayesian

network

The

attribute

sets

whose

marginal

distributions

have

been

cached

between

the

two

stages

of

the

algorithm

are

not

counted

twice

The

time

does

not

include

the

initial

run

of

the

Apriori

algorithm

used

to

find

frequent

itemsets

in

the

data

-LRB-

the

time

of

the

AprioriBN

algorithm

is

included

though

-RRB-

The

times

for

larger

networks

can

be

substantial

however

the

proposed

method

has

still

a

huge

advantage

over

manually

evaluating

thousands

of

frequent

patterns

and

there

are

several

possibilities

to

speed

up

the

algorithm

not

yet

implemented

by

the

authors

discussed

in

the

following

section

no

of

marginals

time

-LSB-

s

-RSB-

Figure

Time

of

computation

depending

on

the

number

of

marginal

distributions

computed

for

the

lymphography

database

no

of

attributes

time

-LSB-

s

-RSB-

max

size

max

size

Figure

Time

of

computation

depending

on

the

number

of

attributes

for

datasets

from

Table

The

maximum

interestingness

column

gives

the

interestingness

of

the

most

interesting

attribute

set

found

for

a

given

dataset

It

can

be

seen

that

there

are

still

highly

interesting

patterns

to

be

found

after

using

classical

Bayesian

network

learning

methods

This

proves

that

frequent

pattern

and

association

rule

mining

has

the

capability

to

discover

patterns

which

traditional

methods

might

miss

To

give

a

better

understanding

of

how

the

algorithm

scales

as

the

problem

size

increases

we

present

two

additional

figures

Figure

shows

how

the

computation

time

increases

with

the

number

of

marginal

distributions

that

must

be

computed

from

the

Bayesian

network

It

was

obtained

by

varying

the

maximum

size

of

attribute

sets

between

and

The

value

of

was

used

-LRB-

equivalent

to

one

row

in

the

database

-RRB-

It

can

be

seen

that

the

computation

time

grows

slightly

slower

than

the

number

of

marginal

distributions

The

reason

for

that

is

that

the

more

marginal

distributions

we

need

to

compute

the

more

opportunities

we

have

to

avoid

using

bucket

elimination

by

using

direct

marginalization

from

a

superset

instead

Determining

how

the

computation

time

depends

on

the

Research

Track

Paper

dataset

#attrs

max

size

#marginals

time

-LSB-

s

-RSB-

max

inter

KSL

soybean

soybean

breast-cancer

annealing

annealing

mushroom

mushroom

lymphography

lymphography

splice

Table

Performance

evaluation

of

the

algorithm

for

finding

all

interesting

attribute

sets

size

of

the

network

is

difficult

because

the

time

depends

also

on

the

network

structure

and

the

number

of

marginal

distributions

computed

-LRB-

which

in

turn

depends

on

the

maximum

size

of

attribute

sets

considered

-RRB-

We

nevertheless

show

in

Figure

the

numbers

of

attributes

and

computation

times

plotted

against

each

other

for

some

of

the

datasets

from

Table

Data

corresponding

to

maximum

attribute

set

sizes

equal

to

and

are

plotted

sepa-rately

It

can

be

seen

that

the

algorithm

remains

practically

usable

for

fairly

large

networks

of

up

to

variables

even

though

the

computation

time

grows

exponentially

For

larger

networks

approximate

inference

methods

might

be

necessary

but

this

is

beyond

the

scope

of

this

paper

A

method

of

computing

interestingness

of

itemsets

and

attribute

sets

with

respect

to

background

knowledge

encoded

as

a

Bayesian

network

was

presented

We

built

efficient

algorithms

for

computing

interestingness

of

frequent

itemsets

and

finding

all

attribute

sets

with

given

minimum

interestingness

Experimental

evaluation

proved

the

effectiveness

and

practical

usefulness

of

the

algorithms

for

finding

interesting

unexpected

patterns

An

obvious

direction

for

future

research

is

increasing

efficiency

of

the

algorithms

Partial

solution

would

be

to

rewrite

the

code

in

C

or

to

use

some

off-the-shelf

highly

op-timized

Bayesian

network

library

like

Intel

's

PNL

Another

approach

would

be

to

use

approximate

inference

methods

like

Gibbs

sampling

Adding

or

removing

edges

in

a

Bayesian

network

does

not

always

influence

all

of

its

marginal

distributions

Interactiv-ity

of

network

building

could

be

imporved

by

making

use

of

this

property

Usefulness

of

methods

developed

for

mining

emerging

patterns

-LSB-

-RSB-

especially

using

borders

to

represent

collections

of

itemsets

could

also

be

investigated

Another

interesting

direction

-LRB-

suggested

by

a

reviewer

-RRB-

could

be

to

iteratively

apply

interesting

patterns

to

modify

the

network

structure

until

no

further

improvement

in

the

network

score

can

be

achieved

A

similar

procedure

has

been

used

in

-LSB-

-RSB-

for

background

knowledge

represented

by

rules

It

should

be

noted

however

that

it

might

be

better

to

just

inform

the

user

about

interesting

patterns

and

let

him/her

use

their

experience

to

update

the

network

Manually

up-dated

network

might

better

reflect

causal

relationships

between

attributes

Another

research

area

could

be

evaluating

other

probabilistic

models

such

as

log-linear

models

and

chain

graphs

instead

of

Bayesian

networks

-LSB-

-RSB-

R.

Agrawal

T.

Imielinski

and

A.

Swami

Mining

association

rules

between

sets

of

items

in

large

databases

In

Proc

ACM

SIGMOD

Conference

on

Management

of

Data

pages

Washington

D.C.

-LSB-

-RSB-

American

Heart

Association

Risk

factors

High

blood

cholesterol

and

other

lipids

-LSB-

-RSB-

R.

J.

Bayardo

and

R.

Agrawal

Mining

the

most

interesting

rules

In

Proc

of

the

ACM

SIGKDD

Int

l

Conf

on

Knowledge

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Data

Mining

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August

-LSB-

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Susanne

G.

B

ttcher

and

Claus

Dethlefsen

Deal

A

package

for

learning

bayesian

networks

bottcher

dethlefsen

-LSB-

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Rina

Dechter

Bucket

elimination

A

unifying

framework

for

reasoning

Artificial

Intelligence

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Guozhu

Dong

and

Jinyan

Li

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emerging

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Discovering

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and

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In

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Intl.

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on

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William

DuMouchel

and

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Pregibon

Empirical

bayes

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multi-item

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In

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of

the

Seventh

International

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on

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and

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Gray

Gray

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and

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Ullman

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efficiently

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David

Heckerman

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Research

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Track

Paper

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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