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Variational Inference and Message Passing: Robotics & Vision Example

Explore Probabilistic models, Bayesian inference, Variational Inference, Message Passing with an example in Robotics and Vision. Learn about optimizing Bayesian networks using graphical models.

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Variational Inference and Message Passing: Robotics & Vision Example

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  1. Variational Inference and Variational Message Passing John WinnMicrosoft Research, Cambridge 12th November 2004 Robotics Research Group, University of Oxford

  2. Overview • Probabilistic models & Bayesian inference • Variational Inference • Variational Message Passing • Vision example

  3. Overview • Probabilistic models & Bayesian inference • Variational Inference • Variational Message Passing • Vision example

  4. Object class C P(C) Surface colour Lighting colour L S P(L) P(S|C) Image colour I P(I|L,S) Bayesian networks • Directed graph • Nodes represent variables • Links show dependencies • Conditional distributions at each node • Defines a joint distribution: P(C,L,S,I)=P(L) P(C) P(S|C) P(I|L,S)

  5. Bayesian inference Object class C Observed variables V and hidden variables H. Hidden Surface colour Lighting colour Hidden variables includeparameters and latent variables. L S Learning/inference involves finding: Image colour I P(H1, H2…| V) Observed

  6. Bayesian inference vs. ML/MAP • Consider learning one parameter θ • How should we represent this posterior distribution?

  7. Bayesian inference vs. ML/MAP • Consider learning one parameter θ Maximum of P(V| θ) P(θ) P(V| θ) P(θ) θ θMAP

  8. High probability mass Bayesian inference vs. ML/MAP • Consider learning one parameter θ High probability density P(V| θ) P(θ) θ θMAP

  9. Bayesian inference vs. ML/MAP • Consider learning one parameter θ P(V| θ) P(θ) θ θML Samples

  10. Bayesian inference vs. ML/MAP • Consider learning one parameter θ P(V| θ) P(θ) θ Variational approximation θML

  11. Overview • Probabilistic models & Bayesian inference • Variational Inference • Variational Message Passing • Vision example

  12. Variational Inference (in three easy steps…) • Choose a family of variational distributions Q(H). • Use Kullback-Leibler divergence KL(Q||P) as a measure of ‘distance’ between P(H|V) and Q(H). • Find Q which minimises divergence.

  13. Q Minimising KL(P||Q) Q P KL Divergence Exclusive Minimising KL(Q||P) P Inclusive

  14. Minimising the KL divergence • For arbitrary Q(H) fixed maximise minimise where • We choose a family of Q distributions where L(Q) is tractable to compute.

  15. Minimising the KL divergence KL(Q || P) maximise ln P(V) fixed L(Q)

  16. Minimising the KL divergence KL(Q || P) maximise ln P(V) fixed L(Q)

  17. Minimising the KL divergence KL(Q || P) maximise ln P(V) fixed L(Q)

  18. Minimising the KL divergence KL(Q || P) maximise ln P(V) fixed L(Q)

  19. Minimising the KL divergence KL(Q || P) maximise ln P(V) fixed L(Q)

  20. Optimal solution for one factor given by Factorised Approximation • Assume Q factorises No further assumptions are required!

  21. Example: Univariate Gaussian • Likelihood function • Conjugate prior • Factorized variational distribution

  22. Initial Configuration γ μ

  23. After Updating Q(μ) γ μ

  24. After Updating Q(γ) γ μ

  25. Converged Solution γ μ

  26. Lower Bound for GMM

  27. Variational Equations for GMM

  28. Overview • Probabilistic models & Bayesian inference • Variational Inference • Variational Message Passing • Vision example

  29. Variational Message Passing • VMP makes it easier and quicker to apply factorised variational inference. • VMP carries out variational inference using local computations and message passing on the graphical model. • Modular algorithm allows modifying, extending or combining models.

  30. Local Updates For factorised Q, update for each factor depends only on the Markov blanket: Updates can be carried out locally at each node.

  31. For example, the Gaussian distribution T mg é ù é ù g X m g = + - gm + 2 ln P ( X | , ) ln 0 1 1 ê ú ê ú 2 2 - g p 2 / 2 X 2 ë û ë û VMP I: The Exponential Family • Conditional distributions expressed in exponential family form. = + + T ln P ( X | θ ) θ u ( X ) g ( θ ) f ( X ) ‘natural’ parameter vector sufficient statistics vector

  32. = + + T ln P ( X | θ ) θ u ( X ) g ( θ ) f ( X ) = + + T ln P ( Z | X , Y ) φ ( Y , Z ) u ( X ) g ' ( X ) f ' ( Y , Z ) VMP II: Conjugacy • Parents and children are chosen to be conjugate i.e. same functional form X Y same Z • Examples: • Gaussian for the mean of a Gaussian • Gamma for the precision of a Gaussian • Dirichlet for the parameters of a discrete distribution

  33. = + + T ln P ( X | θ ) θ u ( X ) g ( θ ) f ( X ) • Parent to child (X→Z) = + + T ln P ( Z | X , Y ) φ ( Y , Z ) u ( X ) g ' ( X ) f ' ( Y , Z ) • Child to parent (Z→X) VMP III: Messages • Conditionals • Messages X Y Z

  34. VMP IV: Update • Optimal Q(X) has same form as P(X|θ) but with updated parameter vector θ* Computed from messages from parents • These messages can also be used to calculate the bound on the evidence L(Q) – see Winn & Bishop, 2004.

  35. VMP Example • Learning parameters of a Gaussian from N data points. γ μ mean precision (inverse variance) x data N

  36. VMP Example Message from γ to all x. γ μ need initialQ(γ) x N

  37. VMP Example Messages from each xnto μ. γ μ x N Update Q(μ) parameter vector

  38. VMP Example Message from updated μ to all x. γ μ x N

  39. VMP Example Messages from each xnto γ. γ μ x N Update Q(γ) parameter vector

  40. Features of VMP • Graph does not need to be a tree – it can contain loops (but not cycles). • Flexible message passing schedule – factors can be updated in any order. • Distributions can be discrete or continuous, multivariate, truncated (e.g. rectified Gaussian). • Can have deterministic relationships (A=B+C). • Allows for point estimates e.g. of hyper-parameters

  41. VMP Software: VIBES Free download from vibes.sourceforge.net

  42. Overview • Probabilistic models & Bayesian inference • Variational Inference • Variational Message Passing • Vision example

  43. Flexible sprite model Proposed by Jojic & Frey (2001) Set of images e.g. frames from a video x N

  44. Flexible sprite model f π Sprite appearance and shape x N

  45. Flexible sprite model f π Sprite transform for this image (discretised) T m x Mask for this image N

  46. Flexible sprite model b f π Background T m Noise β x N

  47. VMP π f b T m β x N Winn & Blake (NIPS 2004)

  48. Results of VMP on hand video Original video Learned appearance and mask Learned transforms (i.e. motion)

  49. Conclusions • Variational Message Passing allows approximate Bayesian inference for a wide range of models. • VMP simplifies the construction, testing, extension and comparison of models. • You can try VMP for yourself vibes.sourceforge.net

  50. That’s all folks!

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