Probabilistic Models CS 4100: Artificial Intelligence Bayes Nets - - PowerPoint PPT Presentation

SLIDE 1

CS 4100: Artificial Intelligence Bayes’ Nets

Jan-Willem van de Meent, Northeastern University

[These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http://ai.berkeley.edu.]

Probabilistic Models

• Models describe how (a portion of) the world works

ks

• Mo

Model els ar are e al alway ays simplifi ficat cations

• May not account for every variable
• May not account for all interactions between variables
• “All models are wrong; but some are useful.”

– George E. P. Box

• Wha

What do do we do do with h pr proba babi bilistic mode dels?

• We (or our agents) need to reason about unknown variables, given evidence
• Ex

Exampl ple: explanation (diagnostic reasoning)

• Ex

Exampl ple: prediction (causal reasoning)

• Ex

Exampl ple: value of information

Independence Independence

• Tw

Two

• variables are in

independent if if:

• This says that their joint distribution factors into a

product two simpler distributions

• Another form:
• We write:
• In

Indep epen enden ence ce is a a simplifying model eling as assumption

• Em

Empi pirical jo join int dis istrib ibutio ions: at best “close” to independent

• What could we assume for {W

{Weather, Traffic, Cavity, Toothache}?

SLIDE 2

Example: Independence?

T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 T W P hot sun 0.3 hot rain 0.2 cold sun 0.3 cold rain 0.2 T P hot 0.5 cold 0.5 W P sun 0.6 rain 0.4

Example: Independence

• N fa

fair ir, in independent coin

• in flip

flips:

H 0.5 T 0.5 H 0.5 T 0.5 H 0.5 T 0.5

Conditional Independence

• Jo

Joint distribution: P(T (Toothache, Cavit ity, Catch)

• If

If I h I have a a cavi cavity, t , the p probability t that t the p probe cat catch ches es in in it it do doesn't de depe pend d on whether I have a to tooth thache:

• P(+

P(+cat catch ch | +toothach ache, e, +cavi cavity) y) = P(+ P(+cat catch ch | +cavi cavity)

• The

The sa same ind ndepend ndenc nce hol holds s if I don

• n’t

t have a cavi cavity:

• P(+

P(+cat catch ch | +toothach ache, e, -cavi cavity) ) = P(+ P(+cat catch ch| -cavi cavity)

• Ca

Catch is is condit itio ionally lly in independent of Toot Tootha hache he gi given n Ca Cavit ity:

• P(C

P(Cat atch ch | Toothach ache, e, Cavi avity) y) = P(C P(Cat atch ch | Cavi avity) y)

• Equi

Equivalent nt statement nts:

• P(T

P(Toothach ache e | Cat atch ch , Cavi avity) y) = P(T P(Toothach ache e | Cavi avity) y)

• P(T

P(Toothach ache, e, Cat atch ch | Cavi avity) y) = P(T P(Toothach ache e | Cavi avity) y) P(C P(Cat atch ch | Cavi avity) y)

• One can be derived from the other easily

Conditional Independence

• Un

Uncondit itio ional l (a (abs bsolute) ) inde depe pende dence very rare (wh (why?) ?)

• Co

Condi nditiona nal in independence is is our r most basic ic and ro robust form of kn knowledge about uncertain en environmen ents.

• X

X is is co conditional ally indep epen enden ent of

• f Y gi

given Z if if an and only if:

• r
• r, e

, equivalently, i , if a and o

• nly i

if

SLIDE 3

Conditional Independence

• Wha

What ab about this domai ain:

• Tr

Traffic

• Um

Umbrella lla

• Ra

Raining Tr Traffic ⫫ Um Umbrella lla Tr Traffic ⫫ Um Umbrella lla | Rain inin ing

Conditional Independence

• Wha

What ab about this domai ain:

• Fi

Fire

• Smoke

ke

• Al

Alarm Al Alarm ⫫ Fi Fire | Smoke ke

Conditional Independence and the Chain Rule

• Cha

Chain n rul ule:

• Tr

Trivial decom

• mpos
• siti

tion

• n:
• Wi

With h assum umpt ption n of condi nditiona nal inde ndepe pende ndenc nce:

• Ba

Bayes’ ne nets / / graphical mo models he help us us express cond nditiona nal ind ndepend ndenc nce assum umptions ns

Ghostbusters Chain Rule

• Ea

Each h sens nsor de depe pends nds onl nly

• n
• n whe

here the he ghost ghost is

• Tha

That means, ns, the he two

• se

sensor nsors s ar are e co condition

• nal

ally y indep epen enden ent, , gi given n th the ghost t positi tion

• T: Top square is re

red B: B: Bottom square is re red G: G: Ghost is in the top

• Giv

Givens (assu assumption

• ns):

): P( +g +g ) = = 0. 0.5 P( ( -g g ) = = 0.5 P( +t +t | +g +g ) = = 0.8 P( +t +t | -g g ) = = 0.4 P( +b +b | +g +g ) = = 0.4 P( +b +b | -g g ) = = 0.8

P(T,B,G) = P(G) P(T|G) P(B|G)

T B G P(T,B,G)

+t +b +g 0.16 +t +b

• g

0.16 +t

• b

+g 0.24 +t

• b
• g

0.04

• t

+b +g 0.04

• t

+b

• g

0.24

• t
• b

+g 0.06

• t
• b
• g

0.06

SLIDE 4

Bayes’ Nets: Big Picture Bayes’ Nets: Big Picture

• Tw

Two

• prob
• blems with

th using g fu full jo join int dis istrib ributio ion table les as as ou

• ur pr

proba babi bilistic mode dels:

• Unless there are only a few variables,

the joint is WAY too big to represent explicitly

• Hard to learn (estimate) anything empirically

about more than a few variables at a time

• Ba

Bayes’ ne nets: De Describ ribe comple lex jo join int dis istrib ributio ions (mo models) ) us using ng simple, local distribut utions ns (co conditional al probab abilities es)

• More properly called gr

graphi hical mod

• dels
• We describe how var

variab ables es local cally y inter eract act

• Local interactions chain together to give

global, in indir irect in interactio ions

• For about 10 min, we’ll be vague about

how these interactions are specified

Example Bayes’ Net: Insurance Example Bayes’ Net: Car

SLIDE 5

Graphical Model Notation

• No

Node des: v : variables ( s (with d domains) s)

• Can be assigned (ob
• bse

served)

• r unassigned (unob

unobse served)

• Ar

Arcs: interactions

• Similar to CSP constraints
• Indicate “di

direct influence” between variables

• For

Formally: encode conditional independence (more later)

• Fo

For no now: im imagin ine that arro rrows mean dire irect cau causat ation (in gen ener eral al, they ey don’t! t!)

Example: Coin Flips

• N ind

independ ndent nt coin

• in flip

flips

• No

No int interactions ions between n varia iable les: abs absolute te ind independ ndenc nce

X1 X2 Xn

Example: Traffic

• Va

Variables:

• R:

R: It It r rains

• T: The

There is s traffic

• Mo

Model el 1: in independence

• Why

Why is is an agent usin ing mo model 2 be better?

R T R T

• Mo

Model el 2: ra rain in causes tra raffic ic

• Le

Let’s bui uild a caus usal graphi hical model!

• Va

Variables

• T:

T: Tr Traffic

• R:

R: It It r rains

• L:

L: Low Low pressur ssure

• D:

D: Ro Roof drips

• B:

B: Ba Ballgame

• C:

C: Ca Cavit ity

Example: Traffic II

R T L D B C

SLIDE 6

Example: Alarm Network

• Va

Variables

• B:

B: Bu Burglary

• A:

A: Al Alarm goes off

• M:

M: Ma Mary ry calls

• J:

J: Jo John calls

• E:

E: Earthquake ke!

B A J M E

Bayes’ Net Semantics Bayes’ Net Semantics

• A

A se set o

• f n

nodes, o , one p per v variable X

• A

A di directed, , acy acycl clic c gr graph (DA DAG)

• A

A co conditional al distribution for for each each no node

• A co

collect ection

• n of distributions over X, one for

each assignment to parent variables

• CP

CPT: conditional probability table

• Description of a noisy “cau

causal sal” process

A1 X An

Ba Bayes ne net = To Topology (graph) + Lo Local l Cond ndit itio iona nal l Probabilit ilitie ies

Probabilities in BNs

• Ba

Bayes’ ne nets im implic licit itly ly en enco code e joint di distribu butions

• As a product of local conditional distributions
• To see what probability a BN gives to a full assignment,

multiply all the relevant conditionals together:

• Ex

Exampl ple:

P(+cavity, +catch, -toothache) =

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) = P(+cavity)

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y) P(+catch | +cavity)

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) P(-toothache | +cavity)

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SLIDE 7

Probabilities in BNs

• Why

Why are we gua uarant nteed d tha hat setting ng re result lts in in a pro roper r jo join int dis istrib ributio ion? ?

• Cha

Chain n rul ule (v (valid d for all di distribu butions): ):

• As

Assume me co conditional al indep epen enden ences ces: à Co Cons nseque quenc nce:

• No

Not ev ever ery BN can can rep epres esen ent ev ever ery joint di distribu bution

• The to

topology enforces co condition

• nal

al in independencie ies

On Only ly dis distribu ibutio ions wh whose e var variables iables ar are e ab absolutel ely y indep epen enden ent can can be e rep epres esen ented ed by y a a Ba Bayes’ne net wit with no ar arcs cs.

Example: Coin Flips

h 0.5 t 0.5 h 0.5 t 0.5 h 0.5 t 0.5

X1 X2 Xn P(h, h, t, h) = 0.54

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Example: Traffic

R T

+r 1/4

• r

3/4 +r +t 3/4

• t

1/4

• r

+t 1/2

• t

1/2

Example: Alarm Network

Burglary Earthqk Alarm John calls Mary calls B P(B) +b 0.001

• b

0.999 E P(E) +e 0.002

• e

0.998 B E A P(A|B,E) +b +e +a 0.95 +b +e

• a

0.05 +b

• e

+a 0.94 +b

• e
• a

0.06

• b

+e +a 0.29

• b

+e

• a

0.71

• b
• e

+a 0.001

• b
• e
• a

0.999 A J P(J|A) +a +j 0.9 +a

• j

0.1

• a

+j 0.05

• a
• j

0.95 A M P(M|A) +a +m 0.7 +a

• m

0.3

• a

+m 0.01

• a
• m

0.99

SLIDE 8

Example: Traffic

• Ca

Causa sal d direction

R T

+r 1/4

• r

3/4 +r +t 3/4

• t

1/4

• r

+t 1/2

• t

1/2 +r +t 3/16 +r

• t

1/16

• r

+t 6/16

• r
• t

6/16

Example: Reverse Traffic

• Re

Revers rse causality ty?

T R

+t 9/16

• t

7/16 +t +r 1/3

• r

2/3

• t

+r 1/7

• r

6/7 +r +t 3/16 +r

• t

1/16

• r

+t 6/16

• r
• t

6/16

Ex Exactl tly y th the same me joi

• int

t distr tributi tion

• n as

as in pr previous mode del

Causality?

• Whe

When n Bayes’ ne nets reflect the he true ue caus usal patterns ns:

• Often si

simpler (nodes have fewer parents)

• Often easier to think

k about

• Often easi

easier er to

• el

elici cit from ex exper erts

• BN

BNs ne need no not actua ually be cau causal al

• Sometimes no

no causa usal ne net exi xist sts s over the domain (especially if variables are missing)

• E.g. consider the variables Tr

Traffic and Dr Drip ips

• End up with ar

arrows s that at ref eflect ect co correl elat ation, not

• t cau

causat sation

• n
• Wha

What do do the he arrows really mean?

• Top

Topol

• logy
• gy re

really e encodes c conditi tional in independence (wh (whic ich may or may not refle lect causal l structure)

Bayes’ Nets

• So

So far: ho how a Ba Bayes’ ne net enc ncodes a joint nt distribut ution

• Ne

Next: ho how to ans nswer que ueries about ut tha hat distribut ution

• Tod

Today: :

• First assembled BNs using an intuitive notion
• f conditional independence as cau

causal ality

• Then saw that key property is co

condi ditional al indepen dependen dence ce

• Ma

Main goa goal: answer queries about conditional independence and influence

• Af

After that: ho how to ans nswer num numerical que ueries (i (inference)