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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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Presentation on theme: "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."— Presentation transcript:

1 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

2 Review

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

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

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

6

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

8 Compute Z ABCDEProduct

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

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

11 Variable Elimination and Conditioning Query:

12 Pictorial Depiction of Elimination

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

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

15 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?

16 Log is a Monotonic increasing function See Mathematica Notebook

17 Maximizing a concave function: See plot in Mathematica

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


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