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Factor Graphs, Variable Elimination, MLEs Joseph Gonzalez TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A A

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Review

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Draw the Bayesian Network: A A B B C C D D E E Parameters Binary Variables ABCDEABCDE

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Draw the Factor Graph A A B B C C D D E E f1f2f3f4f6f5 Factors Variables

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Write down the Equation A A B B C C D D E E f1f2f3f4

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Compute Z A A B B C C D D E E f1f2f3f4 What’s wrong with?

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Compute Z ABCDEProduct

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Variable Elimination = Being Clever A A B B C C D D E E f1f2f3f4 DEf4(d,e) Dg1 (d) 0 1

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Variable Elimination = Being Clever A A B B C C D D E E f1f2f3f4 BCf2(b,c) CDf3(c,d) BDg1(b,d)

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Variable Elimination and Conditioning Query:

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Pictorial Depiction of Elimination

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Gaussian Distributions Random variables that approximately have Gaussian distributions: Goto Mathematica

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The Likelihood X X Observe i.i.d. data (independent and identically distributed): Likelihood

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Maximizing the Likelihood of the Data Observe i.i.d. data (independent and identically distributed): Maximizing with respect to µ: This is difficult! What can we do?

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Log is a Monotonic increasing function See Mathematica Notebook

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Maximizing a concave function: See plot in Mathematica

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Concave Functions: Not Concave or Convex and therefore difficult to maximize and minimize Easy to maximize and minimize

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