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![Distribu(on
Approxima(on
Approximate p(x) with q(x), which belongs to exponential family
Such that: q(x) = h(x)g(η )exp{η T u(x)}
KL( p || q) = − ∫ p(x)In
q(x)
dx = − ∫ p(x)Inq(x)dx + ∫ p(x)Inp(x)dx
p(x)
= − ∫ p(x)Ing(η )dx − ∫ p(x)η T u(x) dx + const
= − Ing(η ) − η T Ε p( x ) [u(x)] + const
where const terms are independent of the natural parameter η
Minimize KL( p || q) by setting the gradient with repect to η to zero:
=> −∇Ing(η ) = Ε p( x ) [u(x)]
By leveraging formula (2.226) in PRML:
=> E q( x ) [u(x)] = −∇Ing(η ) = Ε p( x ) [u(x)]](https://image.slidesharecdn.com/expectationpropagationresearchworkshop-131215193620-phpapp02/75/Expectation-propagation-23-2048.jpg)
Uploaded byDong Guo
Expectation propagation
The document discusses the Expectation Propagation (EP) theory and its applications in Bayesian inference, particularly in graphical models and machine learning. It outlines key components of EP, including message passing, factor graphs, and the relationship between variable nodes, and illustrates the method with examples and applications, such as Bayesian click-through rate prediction for Microsoft's Bing. References include essential literature on machine learning and Bayesian methods, including frameworks for Bayesian inference.
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Expectation propagation
- 1. Expecta(on Propaga(on Theory and Applica(on Dong Guo Research Workshop 2013 Hulu Internal See more details in hEp://dongguo.me/blog/2014/01/01/expecta(on-‐propaga(on/ hEp://dongguo.me/blog/2013/12/01/bayesian-‐ctr-‐predic(on-‐for-‐bing/
- 2. Outline • • • • Overview Background Theory Applica(ons
- 3.
- 4. Bayesian Paradigm • Infer posterior distribu(on Prior Posterior Make decision Data Note: figure of LDA is from Wikipedia, and the right figure is from paper ‘Web-‐Scale Bayesian Click-‐Through Rate PredicFon for Sponsored Search AdverFsing in MicrosoI’s Bing Search Engine’
- 5. Bayesian inference methods • Exact inference – Belief propaga(on • Approximate inference – Stochas(c (sampling) – Determinis(c • Assumed density filtering • Expecta(on propaga(on • Varia(onal Bayes
- 6. Message passing • A form of communica(on used in mul(ple domains of computer science – Parallel compu(ng (MPI) – Object-‐oriented programming – Inter-‐process communica(on – Bayesian inference • A family of methods to infer posterior distribu(on
- 7. Expecta(on Propaga(on • Belongs to message passing family • Approximated method (itera(on is needed) • Very popular in Bayesian inference, especially in graphic model
- 8. Researchers • Thomas Minka – EP was proposed in PhD thesis • Kevin p. Murphy – Machine Learning A ProbabilisFc PerspecFve
- 9.
- 10. Background • • • • • • (Truncated) Gaussian Exponen(al family Graphic model Factor graph Belief propaga(on Moment matching
- 11. Gaussian and Truncated Gaussian • Gaussian opera(on is a basis for EP inference – Gaussian +*/ Gaussian – Gaussian integral • Truncated Gaussian is used in many EP applica(ons • See details here
- 12. Exponen(al family distribu(on • Very good summary in Wikipedia q(z) = h(z)g(η )exp{η T u(z)} • Sufficient sta(s(cs of Gaussian distribu(on: (x, x^2) • Typical distribu(on Note: above 4 figures are from Wikipedia
- 13. Graphical Models • Directed graph (Bayesian Network) x1 x2 x4 K P(x) = ∏ p(xk | pak ) k=1 x3 • Undirected graph (Condi(onal Random Field) x1 x2 x4 x3
- 14. Factor graph • Express rela(on between variable nodes explicitly • Rela(on in edge -‐> factor node • Hide the difference of BN and CRF in inference • Make inference more intui(onal x1 x2 x4 x3 x1 fa x2 fc x4 c x3
- 15.
- 16. Belief Propaga(on Overview • Exact Bayesian method to infer marginal distribu(on – ‘sum-‐product’ message passing • Key components – Calculate posterior distribu(on of variable node – Two kinds of messages
- 17. Posterior distribu(on of variable node • Factor graph p(X) = ∏ Fs (s, X s ), for any variable x in the graph s∈ne( x ) p(x) = ∑ p(X) = ∑ Xx ∏ Fs (s, X s ) = X x s∈ne( x ) ∏ ∑ F (x, X ) = ∏ s s∈ne( x ) X s in which µ fs −>x (x) = ∑ Fs (x, X s ) Xs Note: the figure is from book ‘PaMern recogniFon and machine learning’ s s∈ne( x ) µ fs −>x (x)
- 18. Message: factor -‐> variable node • Factor graph µ fs −>x (x) = ∑ ...∑ fs (x, x1 ,..., x M ) x1 xM ∏ xm ∈ne( fs ) x µ xm −> fs (xm ), in which {x1 ,..., x M } is the set of variables on which the factor fs depends Note: the figure is from book ‘PaMern recogniFon and machine learning’
- 19. Message: variable -‐> factor node • Factor graph µ xm −> fs (xm ) = ∏ µ fl −>xm (xm ) l∈ne( xm ) fs Summary: posterior distribuFon is only determined by factors !! Note: the figure is from book ‘PaMern recogniFon and machine learning’
- 20. Whole steps of BP • Steps to calculate posterior distribu(on of given variable node – Step 1: construct factor graph – Step 2: treat the variable node as root, and ini(alize messages sent from leaf nodes – Step 3: leverage the message passing steps recursively un(l the root node receives messages from all of its neighbors – Step 4: get the marginal distribu(on by mul(plying all messages sent in Note: the figures are from book ‘PaMern recogniFon and machine learning’
- 21. BP: example • Infer marginal distribu(on of x_3 • Infer marginal distribu(on of every variables Note: the figures are from book ‘PaMern recogniFon and machine learning’
- 22. Posterior is intractable some(mes • Example – Infer the mean of a Gaussian distribu(on p(x | θ ) = (1− w)N(x | θ , I ) + wN(x | 0,aI ) p(θ ) = N(θ | 0,bI ) – Ad predictor Note: the figure is from book ‘PaMern recogniFon and machine learning’
- 23. Distribu(on Approxima(on Approximatep(x) with q(x), which belongs to exponential family Such that: q(x) = h(x)g(η )exp{η T u(x)} KL( p || q) = − ∫ p(x)In q(x) dx = − ∫ p(x)Inq(x)dx + ∫ p(x)Inp(x)dx p(x) = − ∫ p(x)Ing(η )dx − ∫ p(x)η T u(x) dx + const = − Ing(η ) − η T Ε p( x ) [u(x)] + const where const terms are independent of the natural parameter η Minimize KL( p || q) by setting the gradient with repect to η to zero: => −∇Ing(η ) = Ε p( x ) [u(x)] By leveraging formula (2.226) in PRML: => E q( x ) [u(x)] = −∇Ing(η ) = Ε p( x ) [u(x)]
- 24. Moment matching It'scalled moment matching when q(x) is Gaussian distribution then u(x) = (x, x 2 )T => ∫ q(x)x dx = ∫ p(x)x dx, and ∫ q(x)x 2 dx = ∫ p(x)x 2 dx => meanq( x ) = ∫ q(x)x dx = ∫ p(x)x dx = mean p( x ) , variance q( x ) = ∫ q(x)x 2 dx − (meanq( x ) )2 = ∫ p(x)x 2 dx − (mean p( x ) )2 = variance p( x ) • Moments of a distribu(on k'th moment M = ∫ x f (x)dx b k a k
- 25. EXPECTATION PROPAGATION = Belief Propaga(on + Moment matching?
- 26. Key Idea • Approximate each factor with Gaussian distribu(on • Approximate corresponding factor pairs one by one? • Approximate each factor in turn in the context of all remaining factors (Proposed by Minka) refine factor (θ ) by ensuring q new (θ ) ∝ (θ )q j (θ ) is close with f j (θ )q j (θ ) fj fj q(θ ) in which q (θ ) = f j (θ ) j
- 27. EP: The detail steps 1.Initialize all of the approximating factors i (θ ) f 2.Initialize the posterior approximation by setting : q(θ ) ∝ ∏ i (θ ) f i 3.Until convergence : (a). Choose a fator (θ ) to refine. fj q(θ ) (b). Remove (θ ) from the posterior by division : q j (θ ) = fj f j (θ ) (c). Get the new posterior by settting sufficient statistics of q new f j (θ )q j (θ ) (θ ) equal to those of zj f j (θ )q j (θ ) new 1 (minimize KL( || q (θ ))),in which z j = ∫ f j (θ )q j (θ )dθ , and q new (θ ) = j (θ )q j (θ ) f zj k new (θ ) : (θ ) = k q (θ ) (d). Get the refined factor f j fj q j (θ )
- 28. Example: The cluEer problem • Infer the mean of a Gaussian distribu(on • Want to try MLE, but p(x | θ ) = (1− w)N(x | θ , I ) + wN(x | 0,aI ) p(θ ) = N(θ | 0,bI ) • Approximate with q(θ ) = N(θ | m,vI ), and each factor (θ ) = N(θ | mn ,vn I ) fn – Approximate mixture Gaussian using Gaussian Note: the figure is from book ‘PaMern recogniFon and machine learning’
- 29. Example: The cluEer problem(2) • Approximate complex factor(e.g. mixture Gaussian) with Gaussian fn (θ ) in blue, (θ ) in red, and q n (θ ) in green fn Remember variance of q n (θ ) is usually very small, so (θ ) only need to approximate fn (θ ) in small range fn Note: above 2 figures are from book ‘PaMern recogniFon and machine learning’
- 30. Applica(on: Bayesian CTR predictor for Bing • See the details here – Inference step by step – Make predic(on • Some insights – Variance of each feature increases aker every exposure – Sample with more features will have bigger variance • Independent assump(on for features
- 31. Experimenta(on • Dataset is very Inhomogeneous • Performance Model FTRL OWLQN Ad predictor AUC 0.638 0.641 0.639 – Other metrics • Pros: speed, parameter choice cost, online learning support, interpreta(ve, support add more factors • Cons: sparse • Code
- 32. Application: XBOX skillrating system • See details in P793~798 of Machine Learning A ProbabilisFc PerspecFve Note: the figure is from paper: ‘TrueSkill: A Bayesian Skill RaFng System’
- 33. Apply to all Bayesian models • Infer.net (Microsok/Bishop) – A framework for running Bayesian inference in graphical models – Model-‐based machine learning
- 34. References • Books – Chapter 2/8/10 of PaMern RecogniFon and Machine Learning – Chapter 22 of Machine Learning: A ProbabilisFc PerspecFve • Papers – – – – A family of algorithms for approximate Bayesian inference From belief propagaFon to expectaFon propagaFon TrueSkill: A Bayesian Skill RaFng System Web-‐Scale Bayesian Click-‐Through Rate PredicFon for Sponsored Search AdverFsing in MicrosoI’s Bing Search Engine • Roadmap for EP