代写代考 Probabilistic Reasoning: Bayes Nets

Probabilistic Reasoning: Bayes Nets
CSci 5512: Artificial Intelligence II
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January 27, 2022

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Probabilistic Reasoning: Bayes Nets

Bayesian networks
A simple, graphical notation for conditional independence assertions and hence for compact specification of full join distributions
A set of nodes, one per variable
A directed, acyclic graph (link implies direct influence)
A conditional distribution for each node given its parents
Conditional distributions
For each Xi, P(Xi|Parents(Xi))
In the form of a conditional probability table (CPT)
Distribution of Xi for each combination of parent values
Probabilistic Reasoning: Bayes Nets
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Topology of network encodes conditional independence assertions
Weather is independent of the other variables
Toothache, Catch are conditionally independent given Cavity
Probabilistic Reasoning: Bayes Nets
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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: Burglar, 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
Probabilistic Reasoning: Bayes Nets
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Example (cont.)
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Probabilistic Reasoning: Bayes Nets

Compactness
A CPT for Boolean Xi with k Boolean parents 2k rows for the combinations of parent values Each row requires one number
Each variable has no more than k parents
The complete network requires O(n · 2k ) numbers
Grows linearly with n
Full joint distribution requires O(2n)
Example: Burglary network
Full joint distribution requires 25 − 1 = 31 numbers Bayes net requires 10 numbers
Probabilistic Reasoning: Bayes Nets
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Local semantics
Each node is conditionally independent of its nondescendants given its parents
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Probabilistic Reasoning: Bayes Nets

Markov blanket
Each node is conditionally independent of all others given its Markov blanket, i.e., parents + children + children’s parents
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Probabilistic Reasoning: Bayes Nets

Conditional Independence in BNs
Which BNs support x1 ⊥ x2|x3?
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Probabilistic Reasoning: Bayes Nets

d-separation
Given a set of nodes X, Y, and Z, is X conditionally independent of Y given Z?
Use Bayes net to see whether Z d-separates X and Y d-separation process:
Consider X, Y, and Z and their ancestors
Add links between any unlinked nodes with common child (moral graph)
Replace all directed links with undrected links
If Z blocks all paths between Z and Y then Z d-separates X and Y =⇒ X ⊥ Y|Z (they are conditionally independent)
Probabilistic Reasoning: Bayes Nets
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Conditional Independence in BNs
Which BNs support x1 ⊥ x2|x3
Probabilistic Reasoning: Bayes Nets
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Conditional Independence in BNs
Which BNs support x1 ⊥ x2|x3
For (a), x1, x2 are dependent, x3 is collider
Probabilistic Reasoning: Bayes Nets
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Conditional Independence in BNs
Which BNs support x1 ⊥ x2|x3
For (a), x1, x2 are dependent, x3 is collider For (b) – (d), x1 ⊥ x2|x3
Probabilistic Reasoning: Bayes Nets
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Constructing Bayesian networks
Hard problem in general: Structure learning Choose an ordering of variables X1, . . . , Xn For i = 1, . . . , n
Add Xi to network
Select parents from X1,…,Xi−1 such that
P(Xi |Parents(Xi )) = P(Xi |X1, . . . , Xi−1) This choice of parents guarantees global semantics
P(X1,…,Xn) = 􏰃P(Xi|X1,…,Xi−1)
= 􏰃 P (Xi |Parents (Xi )) i=1
Probabilistic Reasoning: Bayes Nets
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Earthquake
TT TF FT FF
.95 .94 .29 .001
How can we compute P(b|j, m)?
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Probabilistic Reasoning: Bayes Nets

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