MIT6_047f08_lec19_slide19

MIT6_047f08_lec19_slide19 - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 6.047 / 6.878 Computational Biology: Genomes, Networks, Evolution Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Introduction to Bayesian Networks 6.047/6.878 Computational Biology: Genomes, Networks, Evolution
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Overview • We have looked at a number of graphical representations of probability distributions – DAG example: HMM – Undirected graph example: CRF • Today we will look at a very general graphical model representation – Bayesian Networks • One application – modeling gene expression •A v i v R e g e v guest lecture – an extension of this basic idea
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Probabilistic Reconstruction • Expression data gives us information about what genes tend to be expressed with others • In probability terms, information about the joint distribution over gene states X: P(X)=P(X 1 ,X 2 ,X 3 ,X 4 ,…,X m ) Can we model this joint distribution?
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Bayesian Networks • Directed graph encoding joint distribution variables X P(X) = P(X1,X2,X3,…,XN) • Learning approaches • Inference algorithms • Captures information about dependency structure of P(X)
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Example 1 – Icy Roads I Icy Roads W Watson Crashes H Holmes Crashes Causal impact Assume we learn that Watson has crashed Given this causal network, one might fear Holmes has crashed too. Why?
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Example 1 – Icy Roads I Icy Roads W Watson Crashes H Holmes Crashes Now imagine we have learned that roads are not icy We would no longer have an increased fear that Holmes has crashed
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Conditional Independence I Icy Roads W Watson Crashes H Holmes Crashes If we know nothing about I, W and H are dependent If we know I, W and H are conditionally independent No Info on I We Know I
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Conditional Independency • Independence of 2 random variables Conditional independence given a third (, ) () ( ) X Y PXY PX P Y ⊥⇔ = |( , | ) ( | ) ( | ) but ( , ) ( ) ( ) necessarily X YZ P X P XZ P P Y =
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Example 2 – Rain/Sprinkler R Rain W Watson Wet H Holmes Wet S Sprinkler Holmes discovers his house is wet. Could be rain or his sprinkler.
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R Rain W Watson Wet H Holmes Wet S Sprinkler Now imagine Holmes sees Watsons grass is wet Now we conclude it was probably rain And probably not his sprinkler Example 2 – Rain/Sprinkler
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Explaining Away R Rain W Watson Wet H Holmes Wet S Sprinkler Initially we had two explanations for Holmes’ wet grass. But once we had more evidence for R, this explained away H and thus no reason for increase in S
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Conditional Dependence R Rain W Watson Wet H Holmes Wet S Sprinkler If we don’t know H, R and S are … independent But if we know H, R and S are conditionally dependent Don’t know H Know H
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Graph Semantics Y X Z Y X Z Y X Z Serial Diverging Converging Each implies a particular independence relationship Three basic building blocks
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Chain/Linear | XZ | Y Y X Z Y X Z Conditional Independence
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Diverging | XZ | Y Y X Z Y X Z Conditional Independence
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Converging | XZ | X ZY Y X Z Y X Z Conditional Dependence - Explaining Away
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Graph Semantics Y X Z Converging Three basic building blocks A B ….
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This note was uploaded on 09/24/2010 for the course EECS 6.047 / 6. taught by Professor Manoliskellis during the Fall '08 term at MIT.

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MIT6_047f08_lec19_slide19 - MIT OpenCourseWare...

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