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Extending Expectation Propagation for Graphical Models

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Extending Expectation Propagation for Graphical Models

Yuan (Alan) Qi

Joint work with Tom Minka

- 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.

What we want

Current

Techniques

Error

Computational Time

- 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

- 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

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- 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

- 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)

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

Where the leave-one-out approximation is

- 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):

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.

Loopy BP (Factorized EP)

Error

Extended EP ?

Monte Carlo

Computational Time

- 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

Guess the position of an object given noisy observations

Object

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e.g.

(random walk)

want distribution of x’s given y’s

Factorized and Gaussian in x

xt

yt

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

Forward Message

Backward Message

Observation Message

- 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

- 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.

- is an integer valued Poisson variate with mean

- 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

p(xT|y1:T)

xT

Mean

Variance

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

Im

Re

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

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

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

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

- 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.

(Chen, Wang, Liu 2000)

Signal-Noise-Ratio

Signal-Noise-Ratio

EP outperforms particle smoothers in efficiency with comparable accuracy.

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The information bits et are coded by a convolutional error-correcting encoder.

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

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

Joint distribution is product of pairwise potentials for all edges:

Want to approximate by a simpler distribution

TreeEP

BP

p(x) q(x) Junction tree

p(x) q(x) Junction tree

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

and

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(1) Incorporate edge (x3 x4)

(2) Incorporate edge (x6 x7)

- 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

- 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

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(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)

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

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

- 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 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

- 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.

Extended EP

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

- 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 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

- 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

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

“messages”

- 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

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

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

Gaussian

implies

Gaussian

- EKF, UKF, ADF are all algorithms for:

Linear,

Gaussian

Nonlinear,

Non-Gaussian

- 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)

- Prediction:
- Measurement:
- Smoothing:

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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