Bayesian networks

1
Bayesian networks

• Chapter 14
• Slide Set 2

2
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)

3
Example

• Suppose we choose the ordering M, J, A, B, E
• P(J M) P(J)?

4
Example

• Suppose we choose the ordering M, J, A, B, E
• P(J M) P(J)?No
• P(A J, M) P(A J)? P(A J, M) P(A)?

5
Example

• Suppose we choose the ordering M, J, A, B, E
• P(J M) P(J)?No
• P(A J, M) P(A J)? P(A J, M) P(A)? No
• P(B A, J, M) P(B A)?
• P(B A, J, M) P(B)?

6
Example

• Suppose we choose the ordering M, J, A, B, E
• P(J M) P(J)?No
• P(A J, M) P(A J)? P(A J, M) P(A)? No
• P(B A, J, M) P(B A)? Yes
• P(B A, J, M) P(B)? No
• P(E B, A ,J, M) P(E A)?
• P(E B, A, J, M) P(E A, B)?

7
Example

• Suppose we choose the ordering M, J, A, B, E
• P(J M) P(J)?No
• P(A J, M) P(A J)? P(A J, M) P(A)? No
• P(B A, J, M) P(B A)? Yes
• P(B A, J, M) P(B)? No
• P(E B, A ,J, M) P(E A)? No
• P(E B, A, J, M) P(E A, B)? Yes

8
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
  

9
Using a Bayesian Network

• Suppose you want to calculate
• P(A true, B true, C true, D true)
• P(A true) P(B true A true)
• P(C true B true) P( D true B true)
  
• (0.4)(0.3)(0.1)(0.95)

A
B
C
D
10
Using a Bayesian Network Example

• Using the network in the example, suppose you
  want to calculate
• P(A true, B true, C true, D true)
• P(A true) P(B true A true)
• P(C true B true) P( D true B true)
  
• (0.4)(0.3)(0.1)(0.95)

This is from the graph structure
A
B
These numbers are from the conditional
probability tables
C
D
11
Inference

• Using a Bayesian network to compute probabilities
  is called inference
• In general, inference involves queries of the
  form
• P( X E )

E The evidence variable(s)
X The query variable(s)
12
Inference
HasAnthrax
HasCough
HasFever
HasDifficultyBreathing
HasWideMediastinum

• An example of a query would be
• P( HasAnthrax true HasFever true, HasCough
  true)
• Note Even though HasDifficultyBreathing and
  HasWideMediastinum are in the Bayesian network,
  they are not given values in the query (ie. they
  do not appear either as query variables or
  evidence variables)
• They are treated as unobserved (hidden) variables

13
The Bad News

• Exact inference in BBNs is NP-hard
• Though feasible for singly-connected networks
• But we can achieve significant improvements
  (e.g., variable elimination)
• There are approximate inference techniques which
  are much faster and give fairly good results
• Next class inference

14
Example

• In your local nuclear power plant, there is an
  alarm that senses when a temperature gauge
  exceeds a given threshold. The gauge measures
  the temperature of the core. Consider the
  Boolean variables A (alarm sounds), FA (alarm
  faulty), FG (gauge is faulty), and the
  multivalued variables G (gauge reading) and T
  (actual core temperature.
• The gauge is more likely to fail when the core
  temperature gets too high
• Lets draw the network (in class)

15
Example (cont)

• Suppose
• there are just two possible actual and measured
  temperatures, normal and high
• The prob that the gauge gives the correct temp is
  X when it is working, but Y when it is faulty.
  Give the CPT for G (in class)
• Suppose
• The alarm works correctly unless it is faulty, in
  which case it never sounds. Give the CPT for A
  (in class)

16
Example (cont.)

• Suppose FAfalse FGfalse (the alarm and gauge
  are working properly) and ATrue (and the alarm
  sounds). What is the probability that the
  temperature is too high? (Thigh?) (in class)