This document outlines probabilistic inference in Bayes networks. It begins with a review of probability theory concepts like joint probability, marginal probability, conditional probability, and Bayes' rule. It then discusses probabilistic inference in Bayes networks, including causal/top-down inference using evidence to determine probabilities, diagnostic/bottom-up inference using effects to determine causes, and "explaining away" where additional evidence makes other probabilities less certain. The document also covers uncertain evidence, D-separation to determine conditional independence, and inference techniques in polytrees.

1. Reasoning with Uncertain
Information
Chapter 19.

2. Page 2===
Outline
 Review of Probability Theory
 Probabilistic Inference
 Bayes Networks
 Patterns of Inference in Bayes Networks
 Uncertain Evidence
 D-Separation
 Probabilistic Inference in Polytrees

3. Page 3===
19.1 Review of Probability Theory
 Random variables
 Joint probability
(B (BAT_OK), M (MOVES) , L (LIFTABLE), G
(GUAGE))
Joint
Probability
(True, True, True, True) 0.5686
(True, True, True, False) 0.0299
(True, True, False, True) 0.0135
(True, True, False, False) 0.0007
„ „
Ex.

4. Page 4===
19.1 Review of Probability Theory
 Marginal probability
 Conditional probability
–Ex. The probability that the battery is charged given that
the arm does not move
Ex.

5. Page 5===
19.1 Review of Probability Theory

6. Page 6===
19.1 Review of Probability Theory
 Chain rule
 Bayes’ rule

–Abbreviation for
where

7. Page 7===
19.2 Probabilistic Inference
 The probability some variable Vi has value vi given the evidence
 =e.
p(P,Q,R) 0.3
p(P,Q,¬R) 0.2
p(P, ¬Q,R) 0.2
p(P, ¬Q,¬R) 0.1
p(¬P,Q,R) 0.05
p(¬P, Q, ¬R) 0.1
p(¬P, ¬Q,R) 0.05
p(¬P, ¬Q,¬R) 0.0
   
 
   
 
 
   RpRp
Rp
RQPpRQPp
Rp
RQp
RQp











3.01.02.0
,,),,,
|
   
 
   
 
 
   RpRp
Rp
RQPpRQPp
Rp
RQp
RQp











1.00.01.0
,,),,,
|
 
    1||
75.0|


RQpRQP
RQp

  ？RQp |

8. Page 8===
Statistical Independence
 Conditional independence
 Mutually conditional independence
 Unconditional independence

9. Page 9===
19.3 Bayes Networks
 Directed, acyclic graph (DAG) whose nodes are
labeled by random variables.
 Characteristics of Bayesian networks
–Node Vi is conditionally independent of any subset of nodes
that are not descendents of Vi.
   )(|)(),(| iiiii VPVpVPVAVp 
graphin theofparentsimmediatethe)(
ofsdescendent
notaregraph thatin thenodesofsetany)(
ii
i
i
VVP
V
VA



10. Page 10===
Bayes Networks
 Prior probability
 Conditional probability table (CPT)
   

k
i
iik VPVpVVVp
1
21 )(|,...,,

11. Page 11===
19.3 Bayes Networks

12. Page 12===
Inference in Bayes Networks
 Causal or top-down inference
– Ex. The probability that the arm moves given that the block is
liftable
     
       
       BpLBMpBpLBMp
LBpLBMpLBpLBMp
LBMpLBMpLMp



,|,|
|,||,|
|,|,|
  855.0| LMp

13. Page 13===
Inference in Bayes Networks
 Diagnostic or bottom-up inference
–Using an effect (or symptom) to infer a cause
–Ex. The probability that the block is not liftable given
that the arm does not move.
  9525.0|  LMp (using a causal reasoning)
(Bayes’ rule)( M| L) ( L|) 0.9525*0.3 0.28575
( L| M)=
( M) ( M) ( M)
 

  
  
   
  
( M|L) (L|) 0.0595*0.7 0.03665
(L| M)=
( M) ( M) ( M)
 

  

  
  
( L| M)=0.88632  

14. Page 14===
Inference in Bayes Networks
 Explaining away（辩解）
–¬B explains ¬M, making ¬L less
– certain
     
 
     
 
     
 
88632.0030.0
,
,|
,
|,|
,
|,
,|










MBp
LpBpLBMp
MBp
LpLBpLBMp
MBp
LpLBMp
MBLp (Bayes’ rule)
(def. of conditional prob.)
(structure of the Bayes network)

15. Page 15===
19.5 Uncertain Evidence
 We must be certain about the truth or falsity of the
propositions they represent.
–Each uncertain evidence node should have a child node,
about which we can be certain.
–Ex. Suppose the robot is not certain that its arm did
not move.
• Introducing M’ : “The arm sensor says that the arm moved”
• We can be certain that that proposition is either true or false.
• p(¬L| ¬B, ¬M’) instead of p(¬L| ¬B, ¬M)
–Ex. Suppose we are uncertain about whether or not the
battery is charged.
• Introducing G : “Battery guage”
• p(¬L| ¬G, ¬M’) instead of p(¬L| ¬B, ¬M’)

16. Page 16===
19.6 D-Separation
  d-separates Vi and Vj if for every
undirected path in the Bayes network
between Vi and Vj, there is some
node, Vb, on the path having one of
the following three properties.
–Vb is in , and both arcs on the
path lead out of Vb
–Vb is in , and one arc on the path
leads in to Vb and one arc leads
out.
–Neither Vb nor any descendant of Vb
is in , and both arcs on the path
lead in to Vb.

17. Page 17===
19.6 D-Separation
•I(G,L|B)
•I(G,L)
•I(B,L)

18. Page 18===
Inference in Polytrees
 Polytree
–A DAG for which there is just one path, along arcs in either
direction, between any two nodes in the DAG.

19. Page 19===
 A node is above Q
–The node is connected to Q only through Q’s parents
 A node is below Q
–The node is connected to Q only through Q’s immediate
successors.
 Three types of evidence.
–All evidence nodes are above Q.
–All evidence nodes are below Q.
–There are evidence nodes both above and below Q.
Inference in Polytrees

20. Page 20===
Evidence Above (1)
 Bottom-up recursive algorithm
 Ex. p(Q|P5, P4)
   
   
   
     
     









7,6
7,6
7,6
7,6
7,6
4|75|67,6|
4,5|74,5|67,6|
4,5|7,67,6|
4,5|7,64,5,7,6|
4,5|7,6,4,5|
PP
PP
PP
PP
PP
PPpPPpPPQp
PPPpPPPpPPQp
PPPPpPPQp
PPPPpPPPPQp
PPPPQpPPQp
(Structure of
The Bayes network)
(d-separation)
(d-separation)

21. Page 21===
Evidence Above (2)
 Calculating p(P7|P4) and p(P6|P5)
 Calculating p(P5|P1)
–Evidence is “below”
–Here, we use Bayes’ rule
         
       
 


2,1
3 3
25|12,1|65|6
34,3|74|34,3|74|7
PP
P P
PpPPpPPPpPPp
PpPPPpPPpPPPpPPp
     
 5
11|5
5|1
Pp
PpPPp
PPp 

22. Page 22===
Evidence Below (1)
Top-down recursive algorithm
     
 
   
     QpQPPpQPPkp
QpQPPPPkp
PPPPp
QpQPPPPp
PPPPQp
|11,14|13,12
|11,14,13,12
11,14,13,12
|11,14,13,12
11,14,13,12|



     
   



9
9
|99|13,12
|9,9|13,12|13,12
P
P
QPpPPPp
QppQPPPpQPPp
     
8
8,8|9|9
P
PpQPPpQPp      9|139|129|13,12 PPpPPpPPPp 

23. Page 23===
Evidence Below (2)
     
     



10
10
|1010|1110|14
|1010|11,14|11,14
P
P
QPpPPpPPp
QPpPPPpQPPp
     
     
     
 
   
         1011,10|1511|1011,10|1511|10,15
1111|10,15
10,15
1111|10,15
10,15|11
1111,10|1510|15
10|1510,15|1110|11
1
11
15
PpPPPpPPpPPPpPPPp
PpPPPpk
PPp
PpPPPp
PPPp
PpPPPpPPp
PPpPPPpPPp
P
P









24. Page 24===
Evidence Above and Below
     
 
   
   












||
|,|
|
|,|
,|
2
2
QpQpk
QpQpk
p
QpQp
Qp
 }11,14,13,12{},4,5{| PPPPPPQp
+ -

25. Page 25===
A Numerical Example (1)
     QpQUkpUQp || 
     
       
80.099.08.001.095.0
,|,|
,||


 
RpQRPpRpQRPp
RpQRPpQPp
R
     
       
019.099.001.001.090.0
,|,|
,||


 
RpQRPpRpQRPp
RpQRPpQPp
R
•Diagnostic reasoning

26. Page 26===
A Numerical Example (2)
 Other techniques
– Bucket elimination
– Monte Carlo method
– Clustering
     
60.02.02.08.07.0
2.0|8.0||

 PUpPUpQUp
     
21.098.02.0019.07.0
98.0|019.0||

 PUpPUpQUp
 
 
  13.003.035.4|,35.4
20.095.021.0|
03.005.06.0|



UQpk
kkUQp
kkUQp