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Bayes Nets

Bayes Nets. Rong Jin. O 1. O 3. O 0. O 2. O 4. q 0. q 1. q 2. q 3. q 4. Hidden Markov Model. Inferring from observations ( o i ) to hidden variables ( q i ) This is a general framework for representing and reasoning about uncertainty

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Bayes Nets

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  1. Bayes Nets Rong Jin

  2. O1 O3 O0 O2 O4 q0 q1 q2 q3 q4 Hidden Markov Model • Inferring from observations (oi) to hidden variables (qi) • This is a general framework for representing and reasoning about uncertainty • Representing uncertain information with random variables (nodes) • Representing the relationship between information with conditional probability distribution (directed arcs) • Infer from observation (shadowed nodes) to the hidden variables (circled nodes)

  3. An Example of Bayes Network • S: It is sunny • L: Ali arrives slightly late • O: Slides are put on web late

  4. S O L Bayes Network Example Absence of an arrow: Random S and O are independent. Knowing S will not help predicate O Two arrows into L: L depends on S and O. Knowing S and O will help predicate L.

  5. S O L Inference in Bayes Network • S = 1, O = 0, P(L) = ? • S = 1, P(O) = ?, P(L) = ? • L = 1, P(S) = ?, P(O) = ? • L = 1, S = 1, P(O) = ?

  6. Conditional Independence • Formal definition: A and B are conditional independent given C iff • Different from independence • Example: • A: shoe size • B: glove size • C: heigh • Shoe size is not independent from glove size C B A

  7. S O L Distinguish Two Cases • A: shoe size • B: glove size • C: heigh C B A Given C: A and B are independent Without C: A and B can be dependent • S: It is sunny • L: Ali arrives slightly late • O: Slides are put on web late Without L: S and O are independent Given L: S and O can be dependent

  8. Cloudy Sprinkle Rain WetGrass Another Example for Bayes Nets Inference questions • W=1, P(R) =? • W= 1, P(C) = ? • W= 1, C = 1, P(S) = ?, P(C) = ?, P(S,R) = ?

  9. Bayes Nets Formalized A Bayes net (also called a belief network) is an augmented directed acyclic graph, represented by the pair V , E where: • V is a set of vertices. • E is a set of directed edges joining vertices. No loops of any length are allowed. Each vertex in V contains the following information: • The name of a random variable • A probability distribution table indicating how the probability of this variable’s values depends on all possible combinations of parental values.

  10. Building a Bayes Net • Choose a set of relevant variables. • Choose an ordering for them • Assume they’re called X1 .. Xm (where X1 is the first in the ordering, X1 is the second, etc) • For i = 1 to m: • Add the Xi node to the network • Set Parents(Xi ) to be a minimal subset of {X1…Xi-1} such that we have conditional independence of Xi and all other members of {X1…Xi-1} given Parents(Xi ) • Define the probability table of P(Xi=k Assignments of Parents(Xi ) ).

  11. Example of Building Bayes Nets Suppose we’re building a nuclear power station. There are the following random variables: GRL : Gauge Reads Low. CTL : Core temperature is low. FG : Gauge is faulty. FA : Alarm is faulty AS : Alarm sounds • If alarm working properly, the alarm is meant to sound if the gauge stops reading a low temp. • If gauge working properly, the gauge is meant to read the temp of the core.

  12. CTL GRL AS FA FG Bayes Net for Power Station GRL : Gauge Reads Low. CTL : Core temperature is low. FG : Gauge is faulty. FA : Alarm is faulty AS : Alarm sounds

  13. Inference with Bayes Nets • Key issue: computing joint probability P(X1=x1^ X2=x2^ ….Xn-1=xn-1^ Xn=xn) • Using the conditional independence relations to simplify the computation

  14. Cloudy Sprinkle Rain WetGrass Example for Inference Inference questions • W=1, P(R) =? • W= 1, P(C) = ? • W= 1, C = 1, P(S) = ?, P(C) = ?, P(S,R) = ?

  15. Problem with Inference using Bayes Nets • Inference • Infer from observations EO to unknown variables Eu Suppose you have m binary-valued variables in your Bayes Net and expression Eo mentions k variables. How much work is the above computation?

  16. Problem with Inference using Bayes Nets • General querying of Bayes nets is NP-complete. • Some solutions: • Belief propagation • Take advantage of the structure of Bayes nets • Stochastic simulation • Similar to the sampling approaches for Bayesian average

  17. More Interesting Questions • Learning Bayes nets • Given the topological structure of a Bayes net, learn all the conditional probability tables from examples • Example: Hierarchical mixture model • Learning the topological structure of Bayes net • Very very hard question • Unfortunately, the lecturer does not have enough knowledge to teach you if he wants to !

  18. Cloudy Sprinkle Rain WetGrass Learning Cond. Probabilities in Bayes Nets • Three types of training examples • C, S, R, W • C, R, W • S, C, W • Maximum likelihood approach for estimating the conditional probabilities • EM algorithm for optimization

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