Bruce D’Ambrosio and Robert...

Summer Institute on Probablility in AI 1994

Inference 2 1

Alg

orit

hms

Par

t II

: C

urre

nt D

irec

tion

s

Bru

ce D

’Am

bros

io a

nd R

ober

t F

ung


Inference 2 2

Intr

oduc

tion

lC

ompu

tati

onal

bre

akth

roug

h of

Bay

es N

ets

is e

xplo

itat

ion

of

cond

itio

nal i

ndep

ende

nce

capt

ured

gra

phic

ally

.

lR

esea

rch

dire

ctio

ns•

expl

oit

addi

tion

al s

truc

ture

•pu

sh e

xpre

ssiv

ity

lSt

ruct

ure

expl

oita

tion

•

Noi

sy-O

rs•

Asy

mm

etri

es (

e.g.

, Sim

ilari

ty N

etw

orks

)

•E

ntro

py

lE

xpre

ssiv

ity

•C

onti

nuou

s va

riab

les


Inference 2 3

Out

line

lR

epre

sent

atio

n•

Noi

sy O

r

•A

sym

met

ries

•C

onti

nuou

s V

ars

lA

ppro

xim

atio

n•

Sim

ulat

ion

•Se

arch


Inference 2 4

Noi

sy O

r

lD

istr

ibut

ion

size

is e

xpon

enti

al in

num

ber

of p

aren

ts•

Dif

ficu

lt t

o ac

quir

e•

Exp

ensi

ve t

o co

mpu

te

lN

oisy

Or

inte

ract

ion

mod

el•

Fin

ding

abs

ent

if a

ll pa

rent

s ab

sent

•pa

rent

con

trib

utio

ns in

depe

nden

t

•w

idel

y us

ed f

or m

ulti

-par

ent

inte

ract

ions

.

lR

epre

sent

atio

ns•

ladd

er m

odel

•lo

cal e

xpre

ssio

nsF2

D3

D2

D1

F1

F4

F3

F5 D4


Inference 2 6

Qui

cksc

ore

lB

asic

exp

ress

ion

is e

xpon

enti

al in

cau

ses

lR

earr

ange

pos

teri

or e

xpre

ssio

n fo

r ef

fici

ent

eval

uati

on

(Hec

kerm

an 8

9):

•L

inea

r in

cau

ses

•L

inea

r in

neg

ativ

e fi

ndin

gs•

Exp

onen

tial

in p

osit

ive

find

ings

lD

oesn

’t t

ake

adva

ntag

e of

top

olog

ical

str

uctu

re


Inference 2 7

Qui

cksc

ore

- N

egat

ive

Fin

ding

s

lP

(D1)

= Σ

D2 P

(F1|

D1,

D2)

*P(F

2|D

1,D

2)*P

(D1)

*P(D

2)l

but

F1

nega

tive

, so:

P(F

1|D

1,D

2) =

C(~

f1|D

1)*C

(~f1

|D2)

lth

eref

ore

•P

(D1)

= Σ

D2C

(~f1

|D1)

*C(~

f1|D

2)*C

(~f2

|D1)

*C(~

f2|D

2)*P

(D1)

*P(D

2)

lre

arra

ngin

g:

•P

(D1)

= (

C(~

f1|D

1)*C

(~f2

|D1)

*P(D

1))

* (Σ

D2C

(~f1

|D2)

*C(~

f2|D

2)*P

(D2)

lL

inea

r in

neg

ativ

e fi

ndin

gs a

nd d

isea

ses

F2

D3

D2

D1

F1

F4

F3

F5 D4


Inference 2 8

Qui

cksc

ore

- P

osit

ive

Fin

ding

s

lP

(D1)

= Σ

D2 P

(F1|

D1,

D2)

*P(F

2|D

1,D

2)*P

(D1)

*P(D

2)l

But

F1

Pos

itiv

e, s

o: P

(F1|

D1,

D2)

= 1

- C

(~f1

|D1)

*C(~

f1|D

2)l

Subs

titu

ting

:

•P

(D1)

= Σ

D2(

1 -

C(~

f1|D

1)*C

(~f1

|D2)

) *

(1 -

C(~

f2|D

1)*C

(~f2

|D2)

) *

P(D

1) *

P(D

2)

ldi

stri

buti

ng o

ver

F1,

F2:

• Σ

D2

(1*1

*P(D

1)*P

(D2)

-C

(~f1

|D1)

*C(~

f1|D

2)*1

*P(D

1)*P

(D2)

•

-1*C

(~f2

|D1)

*C(~

f2|D

2)*P

(D1)

*P(D

2)•

+C

(~f1

|D1)

*C(~

f1|D

2))

* C

(~f2

|D1)

*C(~

f2|D

2) *

P(D

1) *

P(D

2))

lre

arra

ngin

g:

•P

(D1)

*ΣD

2P(D

2) -

C(~

f1|D

1)*P

(D1)

*ΣD

2(C

(~f1

|D2)

*P(D

2))

•

- C

(~f2

|D1)

*P(D

1)*Σ

D2(

C(~

f2|D

2)P

(D2)

)

•

+ C

(~f1

|D1)

C(~

f2|D

1)P

(D1)

*ΣD

2(C

(~f1

|D2)

*C(~

f2|D

2)P

(D2)

)


Inference 2 9

Lad

der

Mod

el

lH

ecke

rman

93

lP

(F12 )

= P

(F1|

D1

only

)

lP

(F11

| F12

, D2)

:

• 1

- F

12 pr

esen

t

•P

(F1|

D2

only

) -

F12

abse

nt

F2

D3

D2

D1

F1

F4

F3

F5 D4

F1

F12

F11

F2

F22

F21

D1

D2


Inference 2 10

Lad

der

mod

el, n

egat

ive

evid

ence

lN

egat

ive

evid

ence

is a

sser

ted

at e

ach

Fx n

ode!

•no

te t

his

mea

ns r

e-w

riti

ng c

hild

dis

trib

utio

ns

•P

’(F

11* |

D2)

= P

(F11

* | D

2, F

12* )

lC

lear

ly s

how

s ne

gati

ve f

indi

ng d

ecou

ples

cau

ses.

F1

F12

F11

F2

F22

F21

D1

D2

D2

F1

2 P

(F11

| D

2, F

12 )

A

bs

Pre

s

Ab

s A

bs

1

.0 0

.0P

res

Ab

s

.4

.6

A

bs

Pre

s

0.0

1.0

Pre

s P

res

0

.0 1

.0

D2

P

’(F1

1 * |

D2)

Ab

sA

bs

1.0

Pre

s

.

6


Inference 2 11

Lad

der

mod

el, P

osit

ive

Evi

denc

e

lP

ost

posi

tive

evi

denc

e on

ly a

t te

rmin

al n

odes

lC

lear

ly li

near

in d

isea

ses

F1

F12

F11

F2

F22

F21

D1

D2

F12/F22

D1

D2

F11/F21

F1/F2


Inference 2 12

Loc

al E

xpre

ssio

ns

lSy

mbo

lic a

lgeb

ra c

an f

acto

r dy

nam

ical

lyl

Ord

er in

whi

ch w

e di

stri

bute

ove

r ev

iden

ce m

atte

rs

F2

D3

D2

D1

F1

F4

F3

F5

D4

Cas

e P

os E

v S

avin

g

1

2

9

1

1

224

5

320

1

423

4

523

5

622

4

722

3

824

7

923

3

1019

0


Inference 2 13

Res

earc

h in

Noi

sy O

r

lG

ener

aliz

ed n

oisy

or

(Sri

niva

s 93

)l

Mul

ti-v

alue

lM

ulti

-lev

el n

etw

orks

(C

PC

S, e

xten

ded

set

fact

orin

g)l

Em

bedd

ing

nois

y or

in g

ener

al n

etw

orks


Inference 2 14

Asy

mm

etri

es

lR

epre

sent

atio

n:•

Val

ue d

epen

dent

inde

pend

ence

•P

(Z|x

,Y)

= .3

•C

onti

ngen

t ex

iste

nce

lIn

fere

nce:

•E

ffic

ient

exp

loit

atio

n of

asy

mm

etry

lD

’Am

bros

io 9

1, H

ecke

rman

and

Gei

ger

93, P

oole

94,

She

noy

??, F

ung

and

Shac

hter

, Bar

low

, DA

rel

ated

lIn

fere

ntia

l cos

t/be

nefi

t un

clea

r

Bet

?

Ret

urn

Val

ue


Inference 2 15

Con

tinu

ous

Var

iabl

es

lSi

gnal

to

sym

bol g

ap:

a m

ajor

em

barr

asm

ent

for

AI

lSo

me

succ

ess

in m

ixed

dis

cret

e/co

ntin

uous

bel

ief

nets

lG

ener

al m

odel

s/si

mul

atio

n-ba

sed

(pre

dict

ive)

• D

emos

(H

enri

on)

lE

xact

, lin

ear

wit

h di

scre

te p

rede

cess

ors

(dia

gnos

tic)

•H

ugin

(Je

nsen

, Ole

sen)

•SP

I (C

hang

and

Fun

g)

lC

onti

nuou

s In

flue

nce

Dia

gram

s (K

enle

y, P

olan

d)

Bea

rin

gC

omm

and

Bac

kp

ress

ure

RP

M


Inference 2 16

App

roxi

mat

ion

lE

xact

Inf

eren

ce is

Har

dl

App

roxi

mat

ion

tech

niqu

es:

•A

ppro

xim

ate

repr

esen

tati

on•

Qua

litat

ive

(Gol

dsch

mid

t, P

earl

, Wel

lman

, ...)

•R

educ

ed m

odel

(B

rees

e, H

ecke

rman

, Hor

vitz

)

•Ig

nori

ng w

eak

depe

nden

cies

(Je

nsen

)

•A

ppro

xim

ate

infe

renc

e•

Sim

ulat

ion

(Pea

rl, F

ung,

Peo

t, ..

.)

•Se

arch

(D

’Am

bros

io, H

enri

on, P

oole

)•

Loc

al e

valu

atio

n (D

rape

r)


Inference 2 17

Bac

kwar

d Si

mul

atio

n

oA

new

alg

orit

hm f

amily

for

pro

babi

listi

c in

fere

nce

in B

ayes

ian

Net

wor

ks.

oA

ddre

sses

tw

o lim

itat

ions

of

curr

ent

sim

ulat

ion

met

hods

(for

war

d, m

arko

v bl

anke

t)�

– s

low

con

verg

ence

whe

n fa

ced

wit

h lo

w-l

ikel

ihoo

d ev

iden

ce�

– s

low

con

verg

ence

wit

h de

term

inis

tic

rela

tion

ship

s�

oN

o m

agic

wit

h re

spec

t to

com

plex

ity

resu

lts.


Inference 2 18

For

war

d Sa

mpl

ing

Exa

mpl

e

D1

E1

D2

D3

E2

D4

lP

ossi

ble

Ord

er•

D1,

D2,

D3,

D4

lW

eigh

t•

P(E

1|d1

)*P

(E2|

d3)


Inference 2 19

Bac

kwar

d Si

mul

atio

n: S

impl

e E

xam

ple

P(X

=x0)

/P(X

=x1)

= 1

0-n

P(E

|X=x

0)/P

(E|X

=x1)

= 1

0m

m >

> n

P(X

=x0|

E)/

P(X

=x1|

E)

= 10

m-n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

X E

tria

ls

pos

teri

orP

(x=

0)


Inference 2 20

Bac

kwar

d Si

mul

atio

n: A

lgor

ithm

lO

rder

ing

•A

nod

e m

ust

be in

stan

tiat

ed t

o be

sam

pled

•T

he u

nion

of

the

pred

eces

sors

of

the

node

s in

the

ord

er m

ust

cove

r al

l the

nod

es in

the

net

wor

k

lSa

mpl

ing

•T

akes

pla

ce f

rom

nor

mal

ized

like

lihoo

ds a

nd s

ets

the

pred

eces

sor

valu

es•

For

war

d sa

mpl

ing

is u

sed

for

node

s w

ith

no d

owns

trea

m

evid

ence

lW

eigh

ting

•Im

port

ance

sam

plin

g•

the

rati

o of

the

pri

or p

roba

bilit

y an

d th

e sa

mpl

ing

prob

abili

ty


Inference 2 21

Bac

kwar

d Si

mul

atio

n:

Wei

ghti

ng E

quat

ion

Z(x)

=P(

x i|xPa

(i))

i∈N∏

P(x j|x

Pa(j

))

P(x j|x

Pa(j

))x Pa

(j)∈

XPa

(j)

∑j∈

Nbo∏

P(x)

=P(

x i|xPa

(i))

i∈N∏

P s(x)=

P(x j|x Pa

(j))

P(x j|x Pa

(j))

x Pa(j

)∈X P

a(j)

∑j∈

N bo∏

Z(x)

=P(

x)P s(x

)


Inference 2 22

Bac

kwar

d Si

mul

atio

n: E

xam

ple

lSi

mul

atio

n ou

t fr

om t

he e

vide

nce

lSa

mpl

ing

sets

a n

ode'

s pr

edec

esso

rs

D1

E1

D2

D3

E2

D4

• P

ossi

ble

Ord

ers

° E

2, E

1, D

2',

D4'

° E

1, D

2',

E2,

D4'


Inference 2 23

Dis

cuss

ion

lIm

prov

ed c

onve

rgen

ce in

low

-lik

elih

ood

situ

atio

nsl

Com

puta

tion

al c

osts

of

norm

aliz

atio

n•

cost

s ca

n be

red

uced

thr

ough

: pr

ecom

puta

tion

and

/or

cach

ing

lIn

tegr

atio

n of

bac

kwar

ds a

nd f

orw

ards

sam

plin

gl

Gro

upin

g of

nod

es f

or s

ampl

ing

lD

ynam

ic n

ode

orde

ring

lH

andl

ing

cont

inuo

us d

eter

min

isti

c re

lati

ons

thro

ugh

func

tion

in

vers

ion.


Inference 2 24

Bac

kwar

d Si

mul

atio

n: S

tatu

s

lH

ave

“van

illa”

impl

emen

tati

on w

orki

ng in

ID

EA

Ll

Hav

e w

orke

d ou

t Q

MR

(e.

g., n

oisy

-or)

for

mul

atio

nl

Cur

rent

ly w

orki

ng o

n Q

MR

impl

emen

tati

on


Inference 2 25

Inst

anti

atio

n C

ontr

ol

lW

hy s

tric

tly

forw

ard

or b

ackw

ard?

lP

arti

al in

stan

tiat

ion

mak

es d

istr

ibut

ion

“inf

orm

ativ

e”l

Ent

ropy

and

oth

er in

form

atio

n-th

eore

tic

mea

sure

s l

Stat

ic v

s D

ynam

ic s

elec

tion


Inference 2 26

Sear

ch

lG

oal:

est

imat

e qu

ery

by c

ompu

ting

mas

s of

larg

e in

stan

tiat

ions

:•

find

inst

anti

atio

n w

ith

max

imum

a-p

oste

rior

i pro

babi

lity.

•re

peat

unt

ill s

uffi

cien

t m

ass

com

pute

d.

lM

etho

d: I

ncre

men

tally

ext

end

an in

stan

tiat

ion

usin

g lo

cal

sear

ch.

lD

’Am

bros

io:

incr

emen

tal p

roba

bilis

tic

infe

renc

e

lH

enri

on:

sear

ch in

larg

e B

N2O

net

sl

Poo

le:

use

of c

onfl

ict

sets

in s

earc

h


Inference 2 28

Sear

ch

lL

ike

sim

ulat

ion,

inst

anti

ate

the

vari

able

s in

one

dis

trib

utio

n at

a t

ime

lU

se h

euri

stic

sea

rch

tech

niqu

es t

o gu

ide

inst

anti

atio

n pr

oces

s

lW

hy?

•D

irec

t so

luti

on o

f M

LC

H (

adm

issa

ble

sear

ch r

equi

red!

)•

Few

larg

e te

rms

mig

ht b

e in

form

ativ

e•

skew

ed -

> n+

1 sc

enar

ios

cont

ain

2/e

mas

s!

lH

ow?

•f(

i) =

g(i

) *

h(i)

•g(

i) -

like

lihoo

d of

par

tial

sce

nari

o•

h(i)

- h

euri

stic

est

imat

e of

max

pro

b of

rem

anin

g va

rs g

iven

in

stan

tiat

ion

so f

ar.

lB

ut:

perf

orm

ance

can

be

poor

•L

arge

sea

rch

spac

e•

exte

nsiv

e ba

cktr

acki

ng


Inference 2 29

Sear

ch -

fir

st e

xam

ple

lM

odel

: A

, C, A

’, C

’, A

’’, C

’’ lo

w e

ntro

pyl

Supp

ose

we

know

I -

I’’

all

zero

, O, O

’ ze

ro, O

’’ 1

lSe

arch

sta

rts

wit

h A

, C, A

’, C

’, A

’’, C

’’ a

ll in

hig

h pr

obab

ility

sta

te (

ok)

l0

mas

s w

hen

we

incl

ude

Ol

Now

wha

t?

•P

oor

sear

ch a

rchi

tect

ure

can

yiel

d ex

pone

ntia

l tim

e

I0I1

CAO

I0’

I1’

C’

A’

O’

I0’’

I1’’

C’’

A’’

O’’


Inference 2 30

Poo

le -

Con

flic

ts in

Sea

rch

lC

an d

eriv

e a

conf

lict

invo

lvin

g A

’’ f

rom

fir

st s

earc

h.l

P(V

|O)

:= M

axim

um P

roba

bilit

y of

any

ass

ignm

ent

to

a co

nflic

t co

ntai

ning

var

s V

.

lh(

i) =

Π P

(V|O

) o

ver

all s

ubse

quen

t in

depe

nden

t co

nflic

ts.

lQ

uick

ly r

evea

ls A

, A’

not

wor

th w

orki

ng o

n.l

Poo

le -

UA

I 92

, UA

I-93

I0I1

CAO

I0’

I1’

C’

A’

O’

I0’’

I1’’

C’’

A’’

O’’


Inference 2 31

Con

flic

ts in

Sea

rch

lC

onfl

ict

set:

{C

’, A

’’}

lh

for

any

part

ial t

erm

not

incl

udin

g C

’ or

A’’

is m

ax P

(c’,

a’’

) co

nsis

tent

wit

h ob

s.

lh

for

any

part

ial t

erm

incl

udin

g C

’ or

A’’

is 1

.0

I0I1

CAO

I0’

I1’

C’

A’

O’

I0’’

I1’’

C’’

A’’

O’’


Inference 2 32

D’A

mbr

osio

- f

acto

ring

and

cac

hing

lB

uild

Eva

luat

ion

tree

.l

Top

dow

n, le

ft t

o ri

ght

sear

ch, b

ut c

ache

res

ults

lA

fter

fir

st f

ail,

AN

Y in

stan

tiat

ion

of le

ft s

ubtr

ee w

ith

sam

e C

’ ha

s sa

me

h.

lU

nlik

e P

oole

, fee

ds b

ack

all i

nfo,

not

jus

t co

nflic

ts (

0s)

lB

ut d

oesn

’t g

ener

aliz

e ov

er c

onfl

ict

set

lD

’Am

bros

io:

DX

-92,

UA

I-93

AC

*

*

O

A’

C’

*

*

O’

A’’

C’’

*

*

O’’

*

*


Inference 2 33

Wha

t is

sea

rch

good

for

?

lM

LC

Hl

Pos

teri

or e

stim

atio

nl

Pol

icy

esti

mat

ion

lA

ssum

ptio

ns:

•lo

okin

g fo

r a

few

goo

d in

stan

tiat

ions

•ea

sy t

o fi

nd

•m

ore

is b

ette

r


Inference 2 34

Exp

erim

ents

0.8

90.9

0.9

10.9

20.9

30.9

40.9

50.9

60.9

70.9

8

010

20

30

40

Terms Computed

(MLCH, System OK)


Inference 2 35

Exp

erim

ents

III

0

0.00

2

0.00

4

0.00

6

0.00

8

0.01

0.01

2

05

1015

2025

30

Terms

(MLCH, Single Fault)


Inference 2 36

Cos

t P

er F

ailu

re

12

17

22

27

32

37

42

11

01

00

10

00


Inference 2 37

Sear

ch in

Noi

sy O

R

lT

opN

(H

enri

on, 8

9)l

Aga

in t

he c

ruci

al a

ssum

ptio

n:•

“...o

nly

a ti

ny f

ract

ion

of t

hem

acc

ount

for

mos

t of

the

pr

obab

ility

mas

s.”

lR

(h,F

) =

P(h

|F)/

P(h

0|F)

lM

EP

(d,H

) =

R(H

ud, F

)/R

(H,F

)•

if M

EP

(d,H

) <

1, d

on’t

add

d y

et

lM

EP

(d,H

) >

ME

P(d

,H+)

•if

it is

n’t

wor

th it

to

add

d no

w, i

t ne

ver

will

be

lA

lgor

ithm

:•

star

t w

ith

null

hypo

thes

is•

cons

ider

all

d w

ith

ME

P >

1

•pi

ck b

est

ME

P, a

dd, e

xpan

d re

sult

ing

hypo

thes

is•

loop


Inference 2 38

Sear

ch -

Ope

n is

sues

lP

roba

bilit

y co

mm

unit

y ah

ead

in e

xplo

itin

g ne

twor

k st

ruct

ure

lA

I au

tom

ated

rea

soni

ng c

omm

unit

y ah

ead

in s

uppo

rtin

g dy

nam

ic r

estr

uctu

ring

.

lM

ixed

dis

cret

e/co

ntin

uous

net

wor

ks?

lIn

stan

tiat

ion

orde

r?


Inference 2 39

Sum

mar

y

lR

epre

sent

atio

n•

Noi

sy O

r

•A

sym

met

ries

•C

onti

nuou

s

lA

ppro

xim

atio

n•

Sear

ch•

Sim

ulat

ion

lA

re w

e do

ne?

•br

oad

outl

ine

seem

s un

ders

tood

•lo

ts o

f cl

eanu

p•

hybr

id a

lgor

ithm

s/ar

chit

ectu

res?

•en

gine

erin

g•

expe

rim

enta

tion

•m

inin

g re

al a

pplic

atio

ns (

eg, Q

MR

)

Bruce D’Ambrosio and Robert...

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