probabilistic reasoning; network-based reasoning
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Probabilistic Reasoning;Probabilistic Reasoning;Network-based reasoning Network-based reasoning
Set 7Set 7
ICS 179, Spring ICS 179, Spring 20102010
Chavurah 5/8/2010
Propositional Reasoning
If Alex goes, then Becky goes: If Chris goes, then Alex goes:
Question: Is it possible that Chris goes to
the party but Becky does not?
Example: party problem
BA
A C
e?satisfiabl ,,
theIs
C B, ACBA
theorynalpropositio
= A
= B
= C
= A
Chavurah 5/8/2010
Probabilistic ReasoningParty example: the weather effect
Alex is-likely-to-go in bad weather Chris rarely-goes in bad weather Becky is indifferent but unpredictable
Questions: Given bad weather, which group of
individuals is most likely to show up at the party?
What is the probability that Chris goes to the party but Becky does not?
P(W,A,C,B) = P(B|W) · P(C|W) · P(A|W) · P(W)
P(A,C,B|W=bad) = 0.9 · 0.1 · 0.5
P(A|W=bad)=.9W A
P(C|W=bad)=.1W C
P(B|W=bad)=.5W B
W
P(W)
P(A|W)P(C|W)P(B|W)
B CA
W A P(A|W)
good 0 .01
good 1 .99
bad 0 .1
bad 1 .9
Chavurah 5/8/2010
Mixed Probabilistic and Deterministic networks
P(C|W)P(B|W)
P(W)
P(A|W)
W
B A C
Query:Is it likely that Chris goes to the party if Becky does not but the weather is bad?
PN
CN
Semantics?
Algorithms?),,|,( ACBAbadwBCP
A→B C→A
B A CP(C|W)P(B|W)
P(W)
P(A|W)
W
B A C
A→B C→A
B A C
5
The problem
All men are mortal T
All penguins are birds T
…
Socrates is a man
Men are kind p1
Birds fly p2
T looks like a penguin
Turn key –> car starts P_n
Q: Does T fly?P(Q)?
True propositions
Uncertain propositions
Logic?....but how we handle exceptionsProbability: astronomical
6
12
13
Alpha and beta are events
15
Burglary is independent of EarthquakeBurglary is independent of Earthquake
Earthquake is independent of burglary
20
26
27
28
29
30
31
32
33
34
35
Bayesian Networks: Representation
= P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)
lung Cancer
Smoking
X-ray
Bronchitis
DyspnoeaP(D|C,B)
P(B|S)
P(S)
P(X|C,S)
P(C|S)
P(S, C, B, X, D)
Conditional Independencies Efficient Representation
Θ) (G,BN
CPD: C B D=0 D=10 0 0.1 0.90 1 0.7 0.31 0 0.8 0.21 1 0.9 0.1
Bayesian networks
Chapter 14 , Russel and Norvig
Section 1 – 2
Outline
Syntax Semantics
Example Topology of network encodes conditional
independence assertions:
Weather is independent of the other variables Toothache and Catch are conditionally
independent given Cavity
Example I'm at work, neighbor John calls to say my alarm is ringing, but
neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?
Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
Network topology reflects "causal" knowledge: A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call
Example contd.
Compactness A CPT for Boolean Xi with k Boolean parents has 2k rows for
the combinations of parent values
Each row requires one number p for Xi = true(the number for Xi = false is just 1-p)
If each variable has no more than k parents, the complete network requires O(n · 2k) numbers
I.e., grows linearly with n, vs. O(2n) for the full joint distribution
For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)
SemanticsThe full joint distribution is defined as the product of the
local conditional distributions:P (X1, … ,Xn) = πi = 1 P (Xi | Parents(Xi))
e.g., P(j m a b e)= P (j | a) P (m | a) P (a | b, e) P (b) P (e)
n
Constructing Bayesian networks 1. Choose an ordering of variables X1, … ,Xn
2. For i = 1 to n add Xi to the network select parents from X1, … ,Xi-1 such that
P (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1)
This choice of parents guarantees:P (X1, … ,Xn) = πi =1 P (Xi | X1, … , Xi-1)
= πi =1P (Xi | Parents(Xi))
(by construction)(chain rule)
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)?
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)?P(A | J, M) = P(A | J)? P(A | J, M) = P(A)?
No
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)?P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)? No
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)?P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? YesP(B | A, J, M) = P(B)? NoP(E | B, A ,J, M) = P(E | A)?P(E | B, A, J, M) = P(E | A, B)?
No
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)?P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? NoP(B | A, J, M) = P(B | A)? YesP(B | A, J, M) = P(B)? NoP(E | B, A ,J, M) = P(E | A)? NoP(E | B, A, J, M) = P(E | A, B)? Yes No
Example
Example contd.
Deciding conditional independence is hard in noncausal directions
(Causal models and conditional independence seem hardwired for humans!)
Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed
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