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Javier Turek and Eyal Regev

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Several causes sharing a common effect. Y Z UiUi e+e+ e-e- X Polytrees U1U1 UnUn … … Like a tree, but with multiple parents.

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Link matrix T x|u T x|u contains the conditional probabilities: P(X=x | U 1 =u 1,…,U n =u n ) 1. The table is huge (contains 2 n entries). 2. Who can know such information anyway? You cannot expect to find the P(X | U 1,…,U n ) table. However, you may know how every U i influences X separately.

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The OR-gate Havent done homework Caught cheating Failed an exam More likely! OR

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Inhibitors Failed an exam Caught cheating OR AND Paid the TA Havent done homework AND Re-doing the course

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Inhibitors U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND Inhibitors are independent Associate probability to an inhibitor.

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Noisy OR-Gate U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND

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Noisy OR-Gate U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND TuTu

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Message Passing – updating X Y Z U e+e+ e-e- X

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Y Z U e+e+ W e-e- X

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Message Passing – updating U,Y,Z Y Z U e+e+ e-e- X

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Y Z U e+e+ W e-e- X Message Passing – updating W,U,Y,Z

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Y Z UiUi e+e+ e-e- X Message Passing – many parents Link matrix U1U1 UnUn … …

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Belief Update – Noisy OR-Gate

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Update messages Where The message to the child is the same:

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Example Windows Vista No electricity Virus2K bug Does not start Wrong date Stolen Password Lost data D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

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Example D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4 0.01\0.990.1\0.90.2\0.8 0.8 0.1 0.9 0.20.5 0.1 0.7 0.8 0.2

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Conjunction query Conjunction query q: find the belief that several events happen simultaneously. Applying the chain rule: Product of m belief updates

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Answering a query In our example: Computing P(q):

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Example Update the belief on M 1 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

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Example Update the belief on M 2 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

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Example Update the belief on M 3 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

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Example Update the belief on M 4 D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

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Example And the final solution is…

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Example And the final solution is…

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Thank You!

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