Graphical Models- Inference -

Graphical Models- Inference -

Wolfram Burgard, Luc De Raedt, Kristian Kersting, Bernhard Nebel

Albert-Ludwigs University Freiburg, Germany

PCWP CO

HRBP

HREKG HRSAT

ERRCAUTERHRHISTORY

CATECHOL

SAO2 EXPCO2

ARTCO2

VENTALV

VENTLUNG VENITUBE

DISCONNECT

MINVOLSET

VENTMACHKINKEDTUBEINTUBATIONPULMEMBOLUS

PAP SHUNT

ANAPHYLAXIS

MINOVL

PVSAT

FIO2PRESS

INSUFFANESTHTPR

LVFAILURE

ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME

HYPOVOLEMIA

CVP

BP

Mainly based on F. V. Jensen, „Bayesian Networks and Decision Graphs“, Springer-Verlag New York, 2001.

AdvancedI WS 06/07

Most Probable Explanation

Bayesian Networks

Bayesian Networks

AdvancedI WS 06/07 Outline

• Introduction • Reminder: Probability theory• Basics of Bayesian Networks• Modeling Bayesian networks• Inference (VE, Junction

tree,MPE)• Excourse: Markov Networks• Learning Bayesian networks• Relational Models

Bayesian Networks

Bayesian Networks

AdvancedI WS 06/07

∑∏b

Elimination operator

P(a|e=0)

factor B:

P(a)

P(c|a)

P(b|a) P(d|b,a) P(e|b,c)

facotr C:

factor D:

factor E:

factor A:

e=0

B

C

D

E

A

e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

VE, Bucket elimination [Dechter ‘96]

- Inference (MPE)

- Inference (MPE)

Bayesian Networks

Bayesian Networks

AdvancedI WS 06/07

Finding MPE [Dechter ‘96]

∏b

maxElimination operator

MPE

factor B:

P(a)

P(c|a)

P(b|a) P(d|b,a) P(e|b,c)

factor C:

factor D:

factor E:

factor A:

e=0

B

C

D

E

A

e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

)xP(maxMPEx

=

),|(),|()|()|()(maxby replaced is

,,,,cbePbadPabPacPaPMPE

:

bcdea=

∑ max

- Inference (MPE)

- Inference (MPE)

Bayesian Networks

Bayesian Networks

AdvancedI WS 06/07 Generating the MPE-tuple

C:

E:

P(b|a) P(d|b,a) P(e|b,c)B:

D:

A: P(a)

P(c|a)

e=0 e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

(a)hP(a)max arga' 1. E

a⋅=

0e' 2. =

)e'd,,(a'hmax argd' 3. C

d=

)e'c,,d',(a'h

)a'|P(cmax argc' 4.B

c

××=

)c'b,|P(e')a'b,|P(d')a'|P(bmax argb' 5.

b××

×=

)e',d',c',b',(a' Return

- Inference (MPE)

- Inference (MPE)