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Belief Nets
Notes 6: CDS
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Bayesian Networks
•
Closely related ideas
–
Belief nets
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Influence ets
•
Networks of concepts linked with conditional probabilities
•
Now easy to calculate for large, moderately complex nets
–
Connectivity

maximum number of connected nodes
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Size

total number of nodes
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Sick
Dry
Loses Leav
A Simple Bayesian Network
p(Sick)
= 0.1
p(NotSick)=0.9
p(Dry)
= 0.1
p(NotDry) =0.9
p(LosesLvs Dry,Sick)=0.95
p(NotLosesLvsDry,Sick)=0.05
p(Losesvs Dry, NotSick)=0.85
p(NotLosesLvsDry, NotSick)=0.15
etc.
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Initial Situation
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Probability of ‘Looses Leaves’ set to 100%
Probability of Dry and Sick about equal
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Probability of Dry set to 0
Probability of ‘Sick’ now much higher
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Add some Evidence for being dry
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Initialise the Network
No information really on Leaves:
Just the ‘prior probability on everything
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Now Set LosesLeaves to True
Probability of Dry and Sick both go up,
but Dry goes up much more.
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Now set GrassBrown to ‘yes’
Probability of Dry jumps to 97.95%
but look also at probability of NearbyTree
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Say there is also a nearby tree
Unsurprisingly, the probability of Dry is nearly certain
The probability of Sick drops 10% or so
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Even Consider there is no tree
The evidence of the brown grass is enough to
favour ‘Dry’ over ‘Sick’ by over 4:1
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Evidence that is ‘Explained away’ is not
evidence: consider...
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Losing Leaves, no nearby tree
Sick vs Dry roughly 50:50
(as before)
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Add that the grass is brown
Odds in favour of Dry over Sick rise to 4:1
(84:22)
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Now say that weedkiller was applied
The evidence from GrassBrown is explained away
Odds go back to 50:50
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Diagnosis from Evidence and
Diagnosis by Exclusion
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Previous examples were diagnosis from evidence
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Normally evidence is a manifestation of the problem
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Dryness causes brown grass
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Sometimes we also reason from known causes, e.g.
Nearby trees can add to dryness
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Diagnosis of Exclusion
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If there is evidence against all other causes, then the probability of
what is left must rise
–
Consider the next example...
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Initialise the Network
No information really on Leaves:
Just the ‘prior probability on everything
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Now Set LosesLeaves to True
Probability of Dry and Sick both go up,
but Dry goes up much more.
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Now say there is no nearby tree
Probability of Sick goes up to 50:50
Probability of Dry falls to 50:50
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Now add that the grass is not brown
Probability of Dry falls to under 10%
Probability of Sick rises to 75%
‘Diagnosis of Exclusion’
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Bayesian Nets: Summary so far
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Probabilities propagate
•
Probability must go someplace
–
The good news: Diagnoses of exclusion
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The bad news: spurious conclusions
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IF fungus THEN Black Mould
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if P then Q = not both P and not Q
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P
Q
¬(P & ¬Q)
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Equivalent to p(P&¬Q)
0
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If Black Mould then Fungus
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There is never black mould without fungus
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No fungus implies no black mould
Fungus
yes
no
Black Mould
visible
0.85
0.001
not visible
0.15
0.999
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A More Complex Network Initialised
Sick to Dry roughly 50:50
Note Normal Rainfall
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Set Black Mould
Sick goes to nearly 100% (because of Fungus)
Note that Normal Rain=yes rises
Is this sensible?
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Now also set BrownGrass to yes
Dry goes up to 68%
Normal rain = yes falls to 73% from 89%
Two things can both be wrong at once!
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Note: Black Mould is indirect evidence for
‘Sick’: It is evidence for a mutual cause
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Summary of Bayesian Nets
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Formalism is strictly based on probability
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Normal practice is to use an ‘influence diagram’ and for arrows in
the direction of causation: p(EH)
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Require lots and lots of numbers
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But there are many ways of approximating them
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Allow reasoning
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From cause to effect
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From effect to cause
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By exclusion
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But beware of “residual diagnoses”
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P
Q shown by negation (probability
0)
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equivalences: ¬Q
¬P
¬(P&¬Q)
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Tools
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Several sets of free tools
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Hugin

Lite
(used for this handout)
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www.hugin.com
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Excellent tutorials and documentation on Web
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Belief net site
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http://bayes.stat.washington.edu/almond/belief.html
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GeNIe and SMILE
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http://www2.sis.pitt.edu/~genie
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C++ and graphics packages
»
Good project materal
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