Expectation Propagation for Graphical Models

Expectation Propagation for Graphical Models

Yuan (Alan) Qi

Joint work with Tom Minka

Motivation

• Graphical models are widely used in real-world applications, such as wireless communications and bioinformatics.

• Inference techniques on graphical models often sacrifice efficiency for accuracy or sacrifice accuracy for efficiency.

• Need a new method that better balances the trade-off between accuracy and efficiency.

Motivation

Efficiency

Acc

urac

y

CurrentTechniques

What we want

Outline

• Background

• Expectation Propagation (EP) on dynamic systems– Poisson tracking– Signal detection for wireless communications

• Tree-structured EP on loopy graphs

• Conclusions and future work

Outline

• Background



• Conclusions

Graphical ModelsDirected Undirected

Generative Bayesian networks Boltzman machines

Conditional (Discriminative)

Maximum entropy Markov models

Conditional random fields

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y1 y2

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y1 y2

Inference on Graphical Models

• Bayesian inference techniques:– Belief propagation(BP): Kalman filtering

/smoothing, forward-backward algorithm– Monte Carlo: Particle filter/smoothers,

MCMC• Loopy BP: typically efficient, but not accurate• Monte Carlo: accurate, but often not efficient

Efficiency vs. Accuracy

Efficiency

Acc

urac

y EP ?

BP

MC

Expectation Propagation in a Nutshell

• Approximate a probability distribution by simpler parametric terms:

• Each approximation term lives in an exponential family (e.g. Gaussian)

a

afp )()( xx a

afq )(~

)( xx

)(~xaf

Update Term Approximation

• Iterate the fixed-point equation by moment matching:

))()(~

||)()((minarg \\

)(~

xxxxx

aa

aa

f

qfqfD

a

ab

ba fq )(

~)(\ xx

Where the leave-one-out approximation is

Outline

• Background



• Conclusions

EP on Dynamic SystemsDirected Undirected





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Object Tracking

Guess the position of an object given noisy observations

1y

4y

Object

1x2x

3x

4x

2y

3y

Bayesian Network

ttt νxx 1

noise tt xy

(random walk)e.g.

want distribution of x’s given y’s

x1 x2 xT

y1 y2 yT

Approximation

1

1111 )|()|()|()(),(t

tttt xypxxpxypxpp yx

1

111111 )(~)(~)(~)(~)()(t

tttttttt xoxpxpxoxpq x

Factorized and Gaussian in x

(proportional)

Message Interpretation

)(~)(~)(~)( 11 tttttttt xpxoxpxq

= (forward msg)(observation)(backward msg)

xt

yt

Forward Message

Backward Message

Observation Message

EP on Dynamic Systems• Filtering: t = 1, …, T

– Incorporate forward message

– Initialize observation message

• Smoothing: t = T, …, 1– Incorporate the backward message

– Compute the leave-one-out approximation by dividing out the old observation messages

– Re-approximate the new observation messages

• Re-filtering: t = 1, …, T– Incorporate forward and observation messages

Extension of EP

• Instead of matching moments, use any method for approximate filtering.– Examples: Extended Kalman filter, statistical

linearization, unscented filter

• All methods can be interpreted as finding linear/Gaussian approximations to original terms

Example: Poisson Tracking

• is an integer valued Poisson variate with mean )exp( txty

Poisson Tracking Model

)01.0,(~)|( 11 ttt xNxxp

)100,0(~)( 1 Nxp

!/)exp()|( tx

tttt yexyxyp t

Approximate Observation Message

• is not Gaussian

• Moments of x not analytic

• Two approaches:– Gauss-Hermite quadrature for moments– Statistical linearization instead of moment-

matching

• Both work well

)|()|( tttt yxpxyp

EP Accuracy Improves Significantly in only a few Iterations

Approximate vs. Exact Posterior

EP vs. Monte Carlo: Accuracy

Variance

Mean

Accuracy/Efficiency Tradeoff

EP for Digital Wireless Communication

• Signal detection problem

• Transmitted signal st =

• vary to encode each symbol

• Complex representation:

)sin( ta

iae

Re

Im

),( a

a

Binary Symbols, Gaussian Noise

• Symbols are 1 and –1 (in complex plane)

• Received signal yt =

• Optimal detection is easy

noises

ty

0s 1s

Fading Channel

• Channel systematically changes amplitude and phase:

• changes over time

noise sxy tt

txty

0sxt

1sxt 11sxt

01sxt

1ty

Benchmark: Differential Detection

• Classical technique

• Use previous observation to estimate state

• Binary symbols only

Bayesian network for Signal Detection

x1 x2 xT

y1 y2 yT

s1 s2 sT

On-line EP Joint Signal Detector and Channel Estimation

• Iterate over the last observations

• Observations before act as prior for the current estimation

)( t

Computational Complexity• Expectation propagation O(nLd2)

• Stochastic mixture of Kalman filters O(LMd2)

• Rao-blackwised paricle smoothers O(LMNd2)

n: Number of EP iterations (Typically, 4 or 5)

d: Dimension of the parameter vector

L: Smooth window length

M: Number of samples in filtering

N: Number of samples in smoothing

Experimental Results

EP outperforms particle smoothers in efficiency with comparable accuracy.

(Chen, Wang, Liu 2000)

Bayesian Networks for Adaptive Decoding

x1 x2 xT

y1 y2 yT

e1 e2 eT

The information bits et are coded by a convolutional

error-correcting encoder.

EP Outperforms Viterbi Decoding

Outline

• Background



• Conclusions

EP on Boltzman machinesDirected Undirected





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Inference on Grids

Problem: estimate marginal distributions of the variables indexed by the nodes in a loopy graph, e.g., p(xi), i = 1, . . . , 16.

X1 X2 X3 X4

X5 X6 X7 X8

X9 X10 X11 X12

X13 X14 X15 X16

Boltzmann Machines

4x

2x 3x

1x

a

afp )()( xx

Joint distribution is product of pair potentials:

Want to approximate by a simpler distribution

BP vs. EP

4x

2x 3x

1x

4x

2x 3x

4x

2x 3x

1x1xBP EP

Junction Tree Representation

p(x) q(x) Junction tree

Approximating an Edge by a Tree

24

443

3442

2441

14

21)(

~),(

~),(

~),(

~),(

xf

xxfxxfxxfxxf

a

aaaa

Each potential f a in p is projected onto the tree-structure of q

Correlations are not lost, but projected onto the tree

Moment Matching

• Match single and pairwise marginals of

• Reduces to exact inference on single loops– Use cutset conditioning

4x

2x 3x

1x

4x

2x 3x

1x

and

Local Propagation

• Original EP: globally propagate evidence to the whole tree– Problem: Computationally expensive

• Exploit the junction tree representation: only locally propagate evidence within the minimal subtree that is directly connected to the off-tree edge.– Reduce computational complexity

– Save memory

x5 x7

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Global propagation

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Local propagation

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4-node Graph

TreeEP = the proposed method, BP = loopy belief propagation, GBP = generalized belief propagation on triangles, MF = mean-field, TreeVB =variational tree.

Fully-connected graphs

Results are averaged over 10 graphs with randomly generated potentials

•TreeEP performs the same or better than all other methods in both accuracy and efficiency!

8x8 grids, 10 trials

Method FLOPS Error

Exact 30,000 0

TreeEP 300,000 0.149

BP/double-loop 15,500,000 0.358

GBP 17,500,000 0.003

TreeEP versus BP and GBP

• TreeEP is always more accurate than BP and is often faster

• TreeEP is much more efficient than GBP and more accurate on some problems

• TreeEP converges more often than BP and GBP

Outline

• Background



• Conclusions

Conclusions

• EP algorithms outperform state-of-art inference methods on graphical models in the trade-off between accuracy and efficiency

Efficiency

Acc

urac

y EP

Future Work

• EP is applicable to a wide range of applications

• EP is sensitive to choice of approximation– How to choose an approximation family (e.g.

tree structure) – More flexible approximation: mixture of EP?– Error bound?

Future WorkDirected Undirected





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End

EP versus BP

• EP approximation is in a restricted family, e.g. Gaussian

• EP approximation does not have to be factorized

• EP applies to many more problems– e.g. mixture of discrete/continuous variables

EP versus Monte Carlo

• Monte Carlo is general but expensive

• EP exploits underlying simplicity of the problem if it exists

• Monte Carlo is still needed for complex problems (e.g. large isolated peaks)

• Trick is to know what problem you have

(Loopy) Belief propagation

• Specialize to factorized approximations:

• Minimize KL-divergence = match marginals of (partially factorized) and (fully factorized)– “send messages”

i

iaia xff )(~

)(~x “messages”

)()( \ xx aa qf

)()(~ \ xx a

a qf

Limitation of BP

• If the dynamics or measurements are not linear and Gaussian, the complexity of the posterior increases with the number of measurements

• I.e. BP equations are not “closed”– Beliefs need not stay within a given family

* or any other exponential family

*

Approximate filtering

• Compute a Gaussian belief which approximates the true posterior:

• E.g. Extended Kalman filter, statistical linearization, unscented filter, assumed-density filter

)|()( ttt yxpxq

EP perspective

• Approximate filtering is equivalent to replacing true measurement/dynamics equations with linear/Gaussian equations

)|(

),|()|(

tt

ttttt yxp

yyxpxyp

)|()|(),|( ttttttt yxpxypyyxp

implies

Gaussian

Gaussian

EP perspective

• EKF, UKF, ADF are all algorithms for:

)|( tt xyp

)|( 1tt xxp

Nonlinear,Non-Gaussian

Linear,Gaussian

)|(~tt xyp

)|(~1tt xxp

Terminology

• Filtering: p(xt|y1:t )

• Smoothing: p(xt|y1:t+L ) where L>0

• On-line: old data is discarded (fixed memory)

• Off-line: old data is re-used (unbounded memory)

Kalman filtering / Belief propagation

• Prediction:

• Measurement:

• Smoothing:

111 )|()|()|( ttttttt dxyxpxxpyxp

)|()|(),|( ttttttt yxpxypyyxp

11111 ),|()|()|(),|( ttttttttttt dxyyxpxxpyxpyyxp

Documents

Expectation Propagation for Graphical Models