1 White Parts from: Technical overview for machine-learning researcher – slides from UAI 1999 tutorialTechnical overview for machine-learning researcher – slides from UAI 1999 tutorial Part II

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4 = C t,h Example: for (ht + htthh), we get p(d|m) = 3!2!/6!

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6 Numerical example for the network X 1  X 2 Imaginary sample sizes denoted N’ ijk Data: (true, true) and (true, false)

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8 Used so far Desired

9 How do we assign structure and parameter priors ? Structure priors: Uniform, partial order (allowed/prohibited edges), proportional to similarity to some a priori network.

10 BDe K2K2

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18 Example: Suppose the hyper distribution for (X 1,X 2 ) is Dir( a 00, a 01,a 10, a 11 ). So how to generate parameter priors?

19 Example: Suppose the hyper distribution for (X 1,X 2 ) is Dir( a 00, a 01,a 10, a 11 ) This determines a Dirichlet distribution for the parameters of both directed models.

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21 Summary: Suppose the parameters for (X 1,X 2 ) are distributed Dir( a 00, a 01,a 10, a 11 ). Then, parameters for X 1 are distributed Dir(a 00 +a 01,a 10 +a 11 ). Similarly, parameters for X 2 are distributed Dir(a 00 +a 10,a 01 +a 11 ).

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26 n Example: f(x+y) = f(x) f(y) n Solution: (ln f )`(x+y) = (ln f )`(x) n and so: (ln f )`(x) = constant n Hence: (ln f )(x) = linear function n hence: f(x) = c e ax n Assumptions: Positive everywhere, Differentiable Functional Equations Example

27 The bivariate discrete case

28 The bivariate discrete case

29 The bivariate discrete case

30 The bivariate discrete case

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