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

Review

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

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

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

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

Compute Z ABCDEProduct

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

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

Variable Elimination and Conditioning Query:

Pictorial Depiction of Elimination

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

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

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?

Log is a Monotonic increasing function See Mathematica Notebook

Maximizing a concave function: See plot in Mathematica

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