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Review: Bayesian networks

Review: Bayesian networks. Example: Cloudy, Sprinkler, Rain, Wet Grass. Review: Probabilistic inference. A general scenario: Query variables: X Evidence ( observed ) variables and their values: E = e Unobserved variables: Y

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Review: Bayesian networks

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  1. Review: Bayesian networks • Example: Cloudy, Sprinkler, Rain, Wet Grass

  2. Review: Probabilistic inference • A general scenario: • Query variables:X • Evidence (observed) variables and their values: E = e • Unobserved variables: Y • Inference problem: answer questions about the query variables given the evidence variables • This can be done using theposterior distribution P(X | E = e) • In turn, the posterior needs to be derived fromthe full joint P(X, E, Y) • Since Bayesian networks can afford exponential savings in representing joint distributions, can they afford similar savings for inference?

  3. Bayesian network inference • In full generality, NP-hard • More precisely, #P-hard: equivalent to counting satisfying assignments • We can reduce satisfiability to Bayesian network inference • Decision problem: is P(Y) > 0? C1 C2 C3 G. Cooper, 1990

  4. Bayesian network inference: Big picture • Exact inference is intractable • There exist techniques to speed up computations, but worst-case complexity is still exponential except in some classes of networks • Approximate inference (not covered) • Sampling, variational methods, message passing / belief propagation…

  5. Exact inference example • Query: P(B | j, m) • How to compute this sum efficiently?

  6. Exact inference example

  7. Exact inference with variable elimination • Key idea: compute the results of sub-expressions in a bottom-up way and cache them for later use • Form of dynamic programming • Exponential complexity in general • Polynomial time and space complexity for polytrees: networks at most one undirected path between any two nodes

  8. Representing people

  9. Bayesian network inference: Big picture • In general, exact inference is intractable • Efficient inference via dynamic programming is possible for polytrees • In other practical cases, must resort to approximate methods (not covered in this class) • Sampling, variational methods, message passing / belief propagation…

  10. Parameter learning • Inference problem: given values of evidence variables E = e, answer questions about query variablesX using the posterior P(X| E = e) • Learning problem: estimate the parameters of the probabilistic model P(X | E) given a training sample{(x1,e1), …, (xn,en)}

  11. Parameter learning • Suppose we know the network structure (but not the parameters), and have a training set of complete observations Training set ? ? ? ? ? ? ? ? ?

  12. Parameter learning • Suppose we know the network structure (but not the parameters), and have a training set of complete observations • P(X | Parents(X)) is given by the observed frequencies of the different values of X for each combination of parent values

  13. Parameter learning • Incomplete observations • Expectation maximization (EM) algorithm for dealing with missing data Training set ? ? ? ? ? ? ? ? ?

  14. Parameter learning • What if the network structure is unknown? • Structure learning algorithms exist, but they are pretty complicated… Training set C ? S R W

  15. Summary: Bayesian networks • Structure • Parameters • Inference • Learning

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