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

Part II

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= Ct,h

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

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Numerical example for the network X1 X2

Imaginary sample sizes denoted N’ijk

Data: (true, true) and (true, false)

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Used so far

Desired

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How do we assign structure and parameter priors ?

Structure priors: Uniform, partial order (allowed/prohibited edges), proportional to similarity to some a priori network.

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BDeK2

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Example: Suppose the hyper distribution for (X1,X2) is Dir( a00, a01 ,a10, a11).

So how to generate parameter priors?

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Example: Suppose the hyper distribution for (X1,X2) is Dir( a00, a01 ,a10, a11)This determines a Dirichlet distribution for the parameters of both directed models.

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Summary: Suppose the parameters for (X1,X2) are distributed Dir( a00, a01 ,a10, a11).Then, parameters for X1 are distributed Dir(a00+a01 ,a10+a11).Similarly, parameters for X2 are distributed Dir(a00+a10 ,a01+a11).

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BDe score:

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Example: f(x+y) = f(x) f(y)Example: f(x+y) = f(x) f(y)Solution: (ln f )`(x+y) = (ln f )`(x) Solution: (ln f )`(x+y) = (ln f )`(x) and so: (ln f )`(x) = constantand so: (ln f )`(x) = constantHence: (ln f )(x) = linear functionHence: (ln f )(x) = linear functionhence: f(x) = c ehence: f(x) = c eaxax Assumptions: Positive everywhere, DifferentiableAssumptions: Positive everywhere, Differentiable

Functional Equations Example

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The bivariate discrete case

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The bivariate discrete case

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The bivariate discrete case

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The bivariate discrete case

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