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Extending Expectation Propagation for Graphical ModelsPowerPoint Presentation

Extending Expectation Propagation for Graphical Models

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## PowerPoint Slideshow about ' Extending Expectation Propagation for Graphical Models' - maribel-herandez

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Presentation Transcript

Outline

Outline

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 method that better balances the trade-off between accuracy and efficiency.

Outline

- Background on expectation propagation (EP)
- Extending EP on Bayesian networks for dynamic systems
- Poisson tracking
- Signal detection for wireless communications

- Tree-structured EP on loopy graphs
- Conclusions and future work

Outline

- Background on expectation propagation (EP)
- Extending EP on Bayesian networks for dynamic systems
- Poisson tracking
- Signal detection for wireless communications

- Tree-structured EP on loopy graphs
- Conclusions and future work

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 on general loopy graphs
- Monte Carlo: accurate, but often not efficient

Expectation Propagation in a Nutshell

- Approximate a probability distribution by simpler parametric terms:
For directed graphs:

For undirected graphs:

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

EP in a Nutshell

- The approximate term minimizes the following KL divergence by moment matching:

Where the leave-one-out approximation is

Limitations of Plain EP

- Can be difficult or expensive to analytically compute the needed moments in order to minimize the desired KL divergence.
- Can be expensive to compute and maintain a valid approximation distribution q(x), which is coherent under marginalization.
- Tree-structured q(x):

Three Extensions

1. Instead of choosing the approximate term to minimize the following KL divergence:

use other criteria.

2. Use numerical approximation to compute moments: Quadrature or Monte Carlo.

3. Allow the tree-structured q(x) to be non-coherentduring the iterations. It only needs to be coherent in the end.

Outline

- Background on expectation propagation (EP)
- Extending EP on Bayesian networks for dynamic systems
- Poisson tracking
- Signal detection for wireless communications

- Tree-structured EP on loopy graphs
- Conclusions and future work

Approximation

Factorized and Gaussian in x

xt

yt

Message Interpretation= (forward msg)(observation msg)(backward msg)

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: statistical linearization, unscented Kalman filter (UKF), mixture of Kalman filters
Turn any filtering method into a smoothing method!

- Examples: statistical linearization, unscented Kalman filter (UKF), mixture of Kalman filters
- All methods can be interpreted as finding linear/Gaussian approximations to original terms.

Example: Poisson Tracking

- is an integer valued Poisson variate with mean

Extension of EP: Approximate Observation Message

- is not Gaussian
- Moments of x not analytic
- Two approaches:
- Gauss-Hermite quadrature for moments
- Statistical linearization instead of moment-matching (Turn unscented Kalman filters into a smoothing method)

- Both work well

EP for Digital Wireless Communication

- Signal detection problem
- Transmitted signal st =
- vary to encode each symbol
- Complex representation:

Im

Re

Binary Symbols, Gaussian Noise

- Symbols are 1 and –1 (in complex plane)
- Received signal yt =
- Optimal detection is easy

Fading Channel

- Channel systematically changes amplitude and phase:
- changes over time

Benchmark: Differential Detection

- Classical technique
- Use previous observation to estimate state
- Binary symbols only

Extended-EP Joint Signal Detector and Channel Estimation

- Turn mixture of Kalman filters into a smoothing method
- Smoothing over the last observations
- Observations before act as prior for the current estimation

Computational Complexity

- Expectation propagation O(nLd2)
- Stochastic mixture of Kalman filters O(LMd2)
- Rao-blackwised particle smoothersO(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 (Often larger than 500)
- N: Number of samples in smoothing (Larger than 50)
- EP is about 5,000 times faster than Rao-blackwised particle smoothers.

Experimental Results

(Chen, Wang, Liu 2000)

Signal-Noise-Ratio

Signal-Noise-Ratio

EP outperforms particle smoothers in efficiency with comparable accuracy.

e2

eT

e1

y1

yT

y2

x1

xT

x2

Bayesian Networks for Adaptive DecodingThe information bits et are coded by a convolutional error-correcting encoder.

EP Outperforms Viterbi Decoding

Signal-Noise-Ratio

- Background on expectation propagation (EP)
- Extending EP on Bayesian networks for dynamic systems
- Poisson tracking
- Signal detection for wireless communications

- Tree-structured EP on loopy graphs
- Conclusions and future work

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X11

X12

X13

X14

X15

X16

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

4-node Loopy Graph

Joint distribution is product of pairwise potentials for all edges:

Want to approximate by a simpler distribution

Two Kinds of Edges

- On-tree edges, e.g., (x1,x4):exactly incorporated into the junction tree
- Off-tree edges, e.g., (x1,x2): approximated by projecting them onto the tree structure

KL Minimization

- KL minimization moment matching
- Match single and pairwise marginals of
- Reduces to exact inference on single loops
- Use cutset conditioning

and

x5 x7

x5 x7

x1 x2

x1 x2

x1 x3

x1 x4

x3 x6

x3 x5

x1 x3

x3 x5

x3 x6

x1 x4

x1 x2

x1 x3

x1 x4

x3 x5

x3 x4

x5 x7

x3 x6

x1 x2

x1 x2

x1 x3

x1 x4

x1 x3

x1 x4

x3 x5

x3 x5

x5 x7

x3 x6

x5 x7

x3 x6

x6 x7

Matching Marginals on Graph(1) Incorporate edge (x3 x4)

(2) Incorporate edge (x6 x7)

Drawbacks of Global Propagation

- Update all the cliques even when only incorporating one off-tree edge
- Computationally expensive

- Store each off-tree data message as a whole tree
- Require large memory size

Solution: Local Propagation

- Allow q(x) be non-coherentduring the iterations. It only needs to be coherent in the end.
- Exploit the junction tree representation: only locallypropagate information within the minimal loop (subtree) that is directly connected to the off-tree edge.
- Reduce computational complexity
- Save memory

x1 x2

x5 x7

x1 x2

x1 x2

x5 x7

x5 x7

x1 x2

x1 x3

x1 x4

x3 x5

x1 x3

x3 x6

x3 x6

x3 x5

x1 x3

x1 x4

x1 x4

x3 x5

x3 x6

x1 x3

x1 x4

x3 x4

x3 x5

x3 x5

x5 x7

x3 x6

x6 x7

(1) Incorporate edge(x3 x4)

(2) Propagate evidence

On this simple graph, local propagation runs roughly 2 times faster and uses 2 times less memory to store messages than plain EP

(3) Incorporate edge (x6 x7)

New Interpretation of TreeEP

- Marry EP with Junction algorithm
- Can perform efficiently over hypertrees and hypernodes

4-node Graph

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

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!

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

- Background on expectation propagation (EP)
- Extending EP on Bayesian networks for dynamic systems
- Poisson tracking
- Signal detection for wireless communications

- Tree-structured EP on loopy graphs
- Conclusions and future work

Conclusions

- Extend EP on graphical models:
- Instead of minimizing KL divergence, use other sensible criteria to generate messages. Effectively turn any filtering method into a smoothing method.
- Use quadrature to approximate messages.
- Local propagation to save the computation and memory in tree structured EP.

Conclusions

State-of-art

Techniques

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

Error

Computational Time

Future Work

- More extensions of EP:
- How to choose a sensible approximation family (e.g. which tree structure)
- More flexible approximation: mixture of EP?
- Error bound?
- Bayesian conditional random fields

- More real-world applications

EP versus BP Iterations

- 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 Iterations

- 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 Iterations

- Specialize to factorized approximations:
- Minimize KL-divergence = match marginals of (partially factorized) and (fully factorized)
- “send messages”

“messages”

Limitation of BP Iterations

- 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 Iterations

- Compute a Gaussian belief which approximates the true posterior:
- E.g. Extended Kalman filter, statistical linearization, unscented filter, assumed-density filter

EP perspective Iterations

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

Gaussian

implies

Gaussian

EP perspective Iterations

- EKF, UKF, ADF are all algorithms for:

Linear,

Gaussian

Nonlinear,

Non-Gaussian

Terminology Iterations

- 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 Iterations

- Prediction:
- Measurement:
- Smoothing:

Approximate an Edge by a Tree Iterations

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

Correlations between two nodes are not lost, but projected onto the tree

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