Basic Properties of Belief Networks

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Basic Properties of Belief Networks

• Simplifying Assumption Let X1,,Xn be the
  variables of a belief network and all variables
  have binary states
• P(X1,,Xn) P P(XiParents(Xi))
• P(X1,,Xp-1) P(X1,,Xp-1,Xp) P(X1,,Xp-1,Xp)
• P(XY) a P(X,Y) where a 1/P(Y)
• Remark These 3 equations are sufficient to
  compute any probability in a belief network
  however, using this approach is highly
  inefficient e.g. with n20 computing P(X1X2)
  would require the addition of 218219
  probabilities. Therefore, more efficient ways to
  compute probabilities are needed e.g. if X1 and
  X2 are independent, only P(X1) needs to be
  computed. Another way to speedup computations is
  using probabilities that are already known and no
  not need to be computed and taking advantage of
  the fact that probabilities add up to 1

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Big Sur 2006 Plan
Hurricane Point (580 feet)

• Speed Work1
• March 8 t 3x1M (9)
• March 16 t 3x1M (10)
• March 22 t 2.5M
• March 25 mp 7M (12)
• March 29 8k 8x800
• April 2 ct 4x1M
• April 7 8k 10x800
• April 12 ct 3x1.5M
• April 18 t 2x2.5M (11)
• April 22 ct 2x2800
• April 26 30k 3x1M

• Long Runs and Races
• Nov. 19 11.5M
• Nov. 26 14.3M
• Dec. 4 12M
• Dec. 11 16.5M
• Dec. 18 18M
• Dec. 28 20M
• Jan. 6 13Mhi
• Jan. 14 20M
• Jan. 21 22M
• Jan. 30 20M
• Feb. 2005 kind of sick
• March 5 16
• March 12 19
• March 18 21
• March 30 21
• April 9 19
• April 15 17
• April 30 Big Sur Marathon

Bixby Creek Bridge (1932)
Bixby Creek Bridge (110 feet)
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D-Separation

• Belief Networks abandon the simple independence
  assumptions of naïve Bayesian systems and replace
  them by a more complicated notion of independence
  called d-separation.
• Problem Given evidence involving a set of
  variables E when are two sets of variables X and
  Y of a belief network independent (d-separated)?
• Why is this question important? If X and Y are
  d-separated (given E)
• P(XYE)P(XE)P(YE) and
• P(XEY)P(XE) easy to show how?
• D-separation is used a lot in belief network
  computations (see P(DS1,S2) example to be
  discussed later) particularly to speed up
  belief network computations.

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D-Separation All paths between members of X and
Y must match one of the following 4 patters
Y
X
E(in E, not in E)
(1a)
(1b)
(2)
(3)
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D-Separation
A
D
C
B
E

• a)  Which of the following statements are implied
  by the indicated network structure answer yes
  and no and give a brief reason for your answer!
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• i) P(A,BC) P(AC)P(BC)
•  yes, because
• ii) P(C,ED) P(CD)P(ED)
•  
•  no, because
• iii) P(CA)P(C)
•  no, because

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Compute P(DS1,S2)!!
S1
D
B
S2

• All 3 variables of B have binary states T,F
• P(D) is a short notation for P(DT) and P(S2D)
  is a short notation for P(S2TDF).
• Bs probability tables contain P(D)0.1,
  P(S1D)0.95, P(S2D)0.8, P(S1D)0.2,
  P(S2D)0.2
• Task Compute P(DS1,S2)

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Computing P(DS1,S2)

• P(DS1,S2)P(D)P(S1D)P(S2D)/P(S1,S2) because
  S1D indep S2D
• P(DS1,S2)P(D)P(S1D)P(S2D)/P(S1,S2) S1D
  indep S2D
• (12) 1(P(D)P(S1D)P(S2D)
  P(D)P(S1D)P(S2D))/P(S1,S2)
• P(S1,S2) P(D)P(S1D)P(S2D)
  P(D)P(S1D)P(S2D)g
• P(DS1,S2) a / a b with
• aP(D)P(S1D)P(S2D) and b P(D)P(S1D)P(S2
  D)
• For the example a0.10.950.80.076 and b
  0.90.20.20.036
• P(DS1,S2)0.076/0.1120.678

S1
D
S2
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John/Fred Late Problem

• Assume that John and Fred do not know each other,
  but take the same bus to come to UH to attend the
  same class moreover, both use an alarm clock,
  and will be late to school if their alarm clock
  does not work moreover, if the bus is late, both
  will come late to school. Assume you have to
  model this information using a belief network
  that consists of the following variables that
  have boolean (true,false) states
• John-no-alarm Johns alarm clock does not work
  (short JNOA)
• Fred-no-alarm Freds alarm clock does not work
  (short FNOA)
• Bus-late The bus that John and Fred take is late
  (short BL)
• John-late John is late at school (short JL)
• Fred-late Fred is late at school (short FL)
• Moreover, assume that probability that Johns
  alarm clock does not work is 0.02, the
  probability that Freds alarm clock malfunctions
  is 0.02, and the probability that the bus is late
  is 0.1. You can also assume that Fred and John
  are on time if their alarm clock works and the
  bus they take isnt late.
• a)      Design the structure of a belief network
  that involves the above variables!
• b) Using your results from the previous step,
  compute P(Bus-latetrueFred-latetrue) !

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Solution Student Late Problem Ass4

• P(FBLFL) P(FLFBL)P(FBL)/P(FL)1x0.1/P(FL)0.1/
  0.117910.848
• P(FL)P(FL,WSF, FBL) P(FL,WSF, FBL)
  P(FL,WSF, FBL) P(FL,WSF, FBL)
• P(FL,WSF, FBL) P(FL WSF,FBL)P(FBL,WSF)
  (because WSF and FBL are d-separable giving no
  evidence) P(FL WSF,FBL)P(WSF) P(FBL)
  10.010.1 0.001
• Same for the other 3 formulas in (2)
• P(FL) 10.010.1 10.010.9 10.990.1
  0.010.990.90.001 0.009 0.099
  0.008910.11791

Problem Specification
FBL
FL
MSF
JL
JBL