# Variational Methods for Graphical Models

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```					        CS498-EA
Reasoning in AI
Lecture #14

Professor: Eyal Amir
Fall Semester 2009

* Some slides due to Fei-Fei Li (Stanford U)

1
Summary So Far in Our Class
• We saw motivating applications
• We discussed two methods for
propositional-logical reasoning
• We studied properties of graphical models
of probability distributions
• We learned 2 kinds of probabilistic
inference methods in graphical models
• We examined 2 methods for learning
parameters of graphical models
2
• Variational Approximations
• Models and inference with dynamic
(temporal) systems: logical, probabilistic
• More expressive representations and
inference:
– First-Order Logic (FOL)
– Relational/First-Order Probabilistic Models
– Semantic Web and Description Logics
• Cross-cutting issues
3
Before we Continue…
• Applications of methods we’ve learned
• Review ideas and techniques
• Reinvigorate our search for more
methods…

4
Memories from Lecture 2…
• Applications of reasoning in AI
– Econometrics
– Social Networks
– Verification of Circuits and Programs
– Natural Language Processing
– Robotics
– Vision
– Computer Security

5
Econometrics Example: A Recession
a country
Model of when a bank(b ) goes into
– What is probability of recession,                 m
bankruptcy?

–   Recession: Recession of a country in [0,1]
–   Market[X]: Quarterly market (X) index
–   Loss[X,Y]: Loss of a bank (Y) in a market (X)
6
–   Revenue[Y]: Revenue of a bank (Y)
Experiments

7
Experiments

8
Social Networks
• Example: school friendships and their effects
Friend(A,B)                        Attr(A)                             Measuremt(A)

1
Friend(A,C)            2          Attr(B)                             Measuremt(B)

Friend(B,C)                        Attr(C)                             Measuremt(C)

Pr( f ( A, B), f ( A, C ), f ( B, C ), a( A), a( B), a(C ), m( A), m( B), m(C )) 
1
1 ( f ( A, B), a( A), a( B))   2 ( f ( A, C ), a( A), a(C ))  3 ( f ( B, C ), a ( B), a(C )) 
Z
 4 (a( A), d ( A))  5 (a ( B), d ( B))   6 (a(C ), d (C )) 

f (.,.),a(.),m(.)        shorthand for Friend(., .), Atrr(.), and Measuremt(.)
1...6    potential func-tions
9
f bob; f tom;                    f bob; f lia;    f ann; f lia;                        f bob; f val;
tom    bob                       lia      bob    lia       ann                         val     bob

f bob; f joe;                     f bob; f ann;                                       f ann; f val;                       f lia;     f val;
joe       bob                     ann      bob                                        val        ann                       val         lia

hbob              bbob             bjoe                btom                  bann               blia               bval                 hval

hjoe                                                                                                                                    hlia

htom                                                                                                                  hann
f joe; f ann;                   f joe;   f lia;                    f joe;    f val;
ann     joe                       lia     joe                       val       joe

f joe;    f tom;                    f tom; f lia;                      f tom; f val;                       f tom; f ann;
tom      joe                       lia      tom                       val      tom                         ann        tom

10
Scaling-Up: Computing Pr(f(x,y))
Time vs Number of People

50,000

45,000
Computation Time in Seconds

40,000

35,000

30,000

25,000

20,000

15,000
Figure 5: Computation time for
10,000

5,000

0
0      50,000           100,000       150,000   200,000
Number of People

11
Application: Hardware Verification
x1                    f1                f3
AND            not
x2
f5
f2                      AND
not

OR
x3                                 f4
Question: Can we set this boolean cirtuit to TRUE?

f5(x1,x2,x3) = a function of the input signal

12
Application: Hardware Verification
x1                      f1               f3
AND            not
x2
f5
f2                      AND
not

OR
x3                                  f4         SAT(f5) ?
Question: Can we set this boolean cirtuit to TRUE?

f5(x1,x2,x3) = f3  f4 = f1  (f2  x3) =     M[x1]=FALSE
(x1  x2)  (x2  x3)         M[x2]=FALSE
M[x3]=FALSE13
Hardware Verification
• Questions in logical circuit verification
– Equivalence of circuits
– Arrival of the circuit to a state (required a
temporal model, which gets propositionalized)
– Achieving an output from the circuit

14
Natural-Language Processing
• Logical semantics
• Probabilistic choice between meanings
• Inference over time

15
Vision: Variability within a category
Intrinsic              Deformation

16
Constellation model of
object categories

17
Burl, Leung, Weber, Welling, Fergus, Fei-Fei, Perona, et al.
Goal

18
Goal

19
Burl, Leung, et al. ’96 ’98 Weber, Welling, et al. ’98 ’00, Fergus, et al. ‘03
Goal
• Use prior knowledge of
other objects
• Estimate uncertainties
in models
• Do full Bayesian learning
• Reduce the number of
training examples

20
Variational Approximation Outline
• Motivation
• Outline of the Variational Approximation
approach
• Loopy Belief Propagation
• Variational methodology
– Sequential approach
– Block approach

21
Variational Inference
(in three easy steps…)
1. Choose a family of variational
distributions Q(H).
2. Use Kullback-Leibler divergence
KL(Q||P) as a measure of ‘distance’
between P(H|V) and Q(H).
3. Find Q which minimises divergence.
22

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