(optional)bayesinf05

# (optional)bayesinf05 - Bayesian Networks Independencies and...

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1 Bayesian Networks: Independencies and Inference Scott Davies and Andrew Moore Note to other teachers and users of these slides. Andrew and Scott would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrew’s tutorials: http://www.cs.cmu.edu/~awm/tutorials . Comments and corrections gratefully received. What Independencies does a Bayes Net Model? In order for a Bayesian network to model a probability distribution, the following must be true by definition: Each variable is conditionally independent of all its non- descendants in the graph given the value of all its parents. This implies But what else does it imply? = = n i i i n X parents X P X X P 1 1 )) ( | ( ) ( K

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2 What Independencies does a Bayes Net Model? Example: Z Y X Given Y , does learning the value of Z tell us nothing new about X ? I.e., is P ( X | Y, Z ) equal to P ( X | Y )? Yes. Since we know the value of all of X ’s parents (namely, Y ), and Z is not a descendant of X , X is conditionally independent of Z . Also, since independence is symmetric, P ( Z | Y , X ) = P ( Z | Y ). Quick proof that independence is symmetric Assume: P ( X|Y, Z ) = P ( X|Y ) Then: ) , ( ) ( ) | , ( ) , | ( Y X P Z P Z Y X P Y X Z P = ) ( ) | ( ) ( ) , | ( ) | ( Y P Y X P Z P Z Y X P Z Y P = (Bayes’s Rule) (Chain Rule) (By Assumption) (Bayes’s Rule) ) ( ) | ( ) ( ) | ( ) | ( Y P Y X P Z P Y X P Z Y P = ) | ( ) ( ) ( ) | ( Y Z P Y P Z P Z Y P = =
3 What Independencies does a Bayes Net Model? Let I < X , Y , Z > represent X and Z being conditionally independent given Y . I < X , Y , Z >? Yes, just as in previous example: All X’s parents given, and Z is not a descendant. Y X Z What Independencies does a Bayes Net Model? I < X ,{ U} , Z >? No. I < X ,{ U , V }, Z >? Yes. Maybe I < X , S , Z > iff S acts a cutset between X and Z in an undirected version of the graph…? Z V U X

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4 Things get a little more confusing X has no parents, so we’re know all its parents’ values trivially Z is not a descendant of X So, I < X ,{}, Z >, even though there’s a undirected path from X to Z through an unknown variable Y. What if we do know the value of
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(optional)bayesinf05 - Bayesian Networks Independencies and...

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