Bayesian networks
Chapter 14
Slide Set 2
Constructing Bayesian networks
•
1. Choose an ordering of variables
X
1
, … ,
X
n
•
2. For
i
= 1 to
n
–
add
X
i
to the network
–
–
select parents from
X
1
, … ,X
i

1
such that
P
(X
i
 Parents(X
i
)) =
P
(X
i
 X
1
, ... X
i

1
)
•
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)?
No
P
(A  J, M) =
P
(A  J)
?
P
(A  J, M) =
P
(A)
?
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)
?
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)
?
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
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
•
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)
A
B
C
D
This is from the
graph structure
These numbers are from the
conditional probability tables
11
Inference
•
Using a Bayesian network to compute
probabilities is called inference
•
In general, inference involves queries of the
form:
P( X  E )
X = The query variable(s)
E = The evidence variable(s)
12
Inference
•
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
HasAnthrax
HasCough
HasFever
HasDifficultyBreathing
HasWideMediastinum
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
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
•
Let’s draw the network (in class)
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)
Example (cont.)
•
Suppose FA=false; FG=false (the alarm and gauge are working properly);
and A=True (and the alarm sounds). What is the probability that the
temperature is too high? (T=high?) (in class)
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