Belief Propagation in a Continuous World - PowerPoint Presentation

Slide1

Belief Propagation in a Continuous World

Andrew Frank 11/02/2009Joint work with Alex Ihler and Padhraic Smyth

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.:

A

Slide2

Graphical Models

Nodes represent random variables.Edges represent dependencies.

C

B

A

C

B

A

C

B

A

Slide3

C

E

D

B

A

Markov Random Fields

E

D

B

C

A

D

A

C

E

B

B

 E | C, D

A



C | B

Slide4

Factoring Probability Distributions

Independence relations  factorization

D

C

B

A

p(A,B,C,D) = f(A) f(B) f(C) f(D) f(A,B) f(B,C) f(B,D)

Slide5

Toy Example: A Day in Court

W

A

E

V

A, E, W

є

{“Innocent”, “Guilty”}

V

є

{“Not guilty verdict”, “Guilty verdict”}

I

G

G

I

I

G

Slide6

Inference

Most probable explanation:Marginalization:

Slide7

Iterative Message Updates

x

Slide8

Belief Propagation

W

A

E

V

m

AE

(E)

m

WE

(E)

m

EV

(V)

Slide9

Loopy BP

C

A

B

D

C

A

B

D

Does this work? Does it make any sense?

Slide10

A Variational Perspective

Reformulate the problem:

True distribution, P

“Tractable” distributions

Best tractable

approximation, Q

Find Q to minimize the

divergence

.

Slide11

Desired traits:Simple enough to enable easy computation

Complex enough to represent P Choose an Approximating Family

e.g.

Fully factored:

Structured:

Slide12

Choose a Divergence Measure

Kullback-Liebler divergence:

Alpha divergence:

Common choices:

Slide13

Behavior of α-Divergence

Source: T.

Minka

. Divergence measures and message passing.

Technical Report MSR-TR-2005-173, Microsoft. Research, 2005.

Slide14

Resulting Algorithms

Assuming a fully-factored form of Q, we get…*Mean field, α = 0Belief propagation, α = 1Tree-reweighted BP, α ≥ 1

* By minimizing “local divergence”:

Q(X

1

, X

2

, …,

Xn) = f(X1) f(X2) … f(X

n)

Slide15

Local vs. Global Minimization

Source: T. Minka. Divergence measures and message passing. Technical Report MSR-TR-2005-173, Microsoft. Research, 2005.

Slide16

Applications

Slide17

Sensor Localization

A

B

C

Slide18

Protein Side Chain Placement

RTDCYGN

+

Slide19

Common traits?

?

Continuous state space:

Slide20

Easy Solution: Discretize!

10 bins

10 bins

Domain size:

d = 100

20 bins

20 bins

Domain size:

d = 400

Each message:

O(d

2

)

Slide21

Particle BP

We’d like to pass “continuous messages”…

C

A

B

D

B

m

AB

(B)

1

4

4.2

5

2.5

…

…

…

Instead, pass discrete messages over sets of particles:

{ b

(

i

)

} ~ W

B

(B)

m

AB

({b

(

i

)

})

b

(1)

b

(2)

b

(N)

. . .

Slide22

PBP: Computing the Messages

Re-write as an expectation:

Finite-sample approximation:

Slide23

Choosing“Good” Proposals

C

A

B

D

Proposal should “match” the integrand.

Sample from the belief:

Slide24

Iteratively Refine Particle Sets

(2)

f(

x

s

,

x

t)

Draw a set of particles, {xs(i)} ~ Ws(

xs).Discrete inference over the particle discretization.

Adjust Ws(xs)

(1)

(3)

XsX

t(1)

(3)

Slide25

Benefits of PBP

No distributional assumptions.Easy accuracy/speed trade-off.Relies on an “embedded” discrete algorithm.

Belief propagation, mean field, tree-reweighted BP…

Slide26

Exploring PBP: A Simple Example

x

s

||

x

s

–

x

t

||

Slide27

Continuous Ising Model

Marginals

Approximate

Exact

Mean Field PBP

α = 0

PBP

α = 1

TRW PBP

α = 1.5

* Run with 100 particles per node

Slide28

A Localization Scenario

Slide29

Exact Marginal

Slide30

PBP Marginal

Slide31

Tree-reweighted PBP Marginal

Slide32

Estimating the Partition Function

Mean field provides a lower bound.Tree-reweighted BP provides an upper bound.p(A,B,C,D) = f(A) f(B) f(C) f(D) f(A,B) f(B,C) f(B,D)

Z = f(A) f(B) f(C) f(D) f(A,B) f(B,C) f(B,D)

Slide33

Partition Function Bounds

Slide34

Conclusions

BP and related algorithms are useful!Particle BP let’s you handle continuous RVs.Extensions to BP can work with PBP, too.Thank You!