1 / 27

Approximate Bayesian Inference I:

Pattern Recognition and Machine Learning Chapter 10. Approximate Bayesian Inference I:. Structural Approximations. Falk LIEDER December 2 2010. Introduction Variational Inference Variational Bayes Applications. Statistical Inference. Z. P(Z|X). X. Hidden States.

iona
Télécharger la présentation

Approximate Bayesian Inference I:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Pattern Recognition and Machine Learning • Chapter 10 ApproximateBayesianInference I: StructuralApproximations • FalkLIEDER December 2 2010

  2. IntroductionVariationalInferenceVariational Bayes Applications Statistical Inference Z P(Z|X) X Hidden States Observations Posterior Belief

  3. IntroductionVariationalInferenceVariational Bayes Applications When Do You Need Approximations? The problemwith Bayes theoremisthatitoftenleadstointegralsthatyoudon’tknowhowtosolve. • Noanalyticsolutionfor • Noanalyticsolutionfor • In thediscretecasecomputinghascomplexity • SequentialLearning For Non-Conjugate Priors

  4. IntroductionVariationalInferenceVariational Bayes Applications HowtoApproximate? Samples Structual Approximation Approximation by a Densityof a given Form Evidence /ExpectationsofApproximateDensityare easy tocompute ApproximateDensitybyHistogram Approx. Expectationsby Averages Numericalintegration Approximate Integrals Numerically: • Evidence p(x) • Expectations Infeasibleif Z is high-dimensional

  5. IntroductionVariationalInferenceVariational Bayes Applications HowtoApproximate? StructuralApproximations (VariationalInference) StochasticApproximations (Monte-Carlo-Methods, Sampling) - Time-Intensive + AsymptoticallyExact - Storage Intensive + EasilyApplicable General PurposeAlgorithms + Fast toCompute • Systematic Error + EfficientRepresentation - Applicationoftenrequiresmathematicalderivations + Learning Rules give Insight

  6. IntroductionVariationalInferenceVariational Bayes Applications VariationalInference—An Intuition Target Family KL-Divergence True Posterior VB Approximation ProbabilityDistributions

  7. IntroductionVariationalInferenceVariational Bayes Applications WhatDoesClosestMean? Intuition: Closestmeans minimal additional surprise on average. Kullback-Leibler (KL) divergencemeasuresaverage additional surprise. KL[p||q] measureshowmuchlessaccuratethe belief q isthan p, if p isthetrue belief. KL[p||q]islargestreduction in averagesurprisethatyoucanachieve, ifp isthetrue belief.

  8. IntroductionVariationalInferenceVariational Bayes Applications KL-Divergence Illustration

  9. IntroductionVariationalInferenceVariational Bayes Applications Properties ofthe KL-Divergence • Zero iffbothargumentsareidentical: • Greaterthanzero, iftheyare different: Disadvantage The KL-divergenceis not a metric (distancefunction), because Itis not symmetric . Itdoes not satisfythetriangleinequality.

  10. IntroductionVariationalInferenceVariational Bayes Applications Howto Find theClosest Target Density? • Intuition: MinimizeDistance • Implementations: • Variational Bayes:Minimize • Expectation Propagation: • Arbitrariness • Different Measures Different Algorithms & Different Results • Alternative Schemesarebeingdeveloped,e.g. Jaakola-Jordan variationalmethod, Kikuchi-Approximations

  11. IntroductionVariationalInferenceVariational Bayes Applications Minimizing Functionals • KL-divergenceis a functional

  12. IntroductionVariationalInferenceVariational BayesApplications VB andthe Free-Energy Variational Bayes: ior Problem: Youcan’tevaluatethe KL-divergence, becauseyoucan’tevaluatethe posterior. Solution: Conclusion: • Youcanmaximizethefree-energyinstead. const

  13. IntroductionVariationalInferenceVariational BayesExamples VB: Minimizing KL-DivergenceisequivalenttoMaximizing Free-Energy (q)

  14. IntroductionVariationalInferenceVariational BayesApplications Constrained Free-EnergyMaximization (q) Intuition: • Maximize a LowerBound on the Log Model Evidence • Maximizationisrestrictedtotractabletargetdensities Definition: Properties • The free-energyis maximal forthetrueposterior.

  15. IntroductionVariationalInferenceVariational BayesApplications VariationalApproximations • FactorialApproximations (Meanfield) • Independence Assumption • Optimizationwithrespecttofactordensities • NoRestriction on Functional Form ofthefactors • Approximation byParametricDistributions • Optimization w.r.t. Parameters • VariationalApproximationsfor Model Comparison • Variational Approximation ofthe Log Model Evidence

  16. IntroductionVariationalInferenceVariational BayesExamples Meanfield Approximation Goal: Rewrite as a function of and optimize. Optimize separately for each factor Step 1:

  17. IntroductionVariationalInferenceVariational BayesApplications Meanfield Approximation, Step 1

  18. IntroductionVariationalInferenceVariational BayesApplications Meanfield Approximation, Step 2 Noticethat. The constant must betheevidence, becausehastointegratetoone. Hence,

  19. IntroductionVariationalInferenceVariational Bayes Applications MeanfieldExample True Distribution: with Target Family: VB meanfieldsolution: • +const • Hence, and • Bysymmetry

  20. IntroductionVariationalInferenceVariational Bayes Applications MeanfieldExample Observation: VB-Approximation ismorecompactthantruedensity. Reason: KL[q||p] does not penalizedeviationswhere q iscloseto 0. True Density Approximation UnreasonableAssumptions Poor Approximation

  21. IntroductionVariationalInferenceVariational Bayes Applications KL[q||p] vs. KL[p||q] Variational Bayes • AnalyticallyEasier • Approx. ismorecompact Expectation Propagation • More Involved • Approx. is wider

  22. IntroductionVariationalInferenceVariational Bayes Applications 2. ParametricApproximations • Problem: • Youdon’tknowhowtointegratepriortimeslikelihood. • Solution: • Approximate by . • KL-divergence and free-energy become functions of the parameters • Applystandardoptimizationtechniques. • Setting derivatives tozero Oneequation per parameter. • Solve System ofEquationsby iterative Updating.

  23. IntroductionVariationalInferenceVariational Bayes Applications Parametric Approximation Example Z Goal: LearntheRewardProbability p • Likelihood: , • Prior: • Posterior: Problem Youcannotderive a learningrulefortheexpectedrewardand ist variance, because… • NoAnalyticFormulaforExpectedRewardProbability • Form of Prior Changeswith Every Observation Solution: Approximatethe Posterior by a Gaussian. X

  24. IntroductionVariationalInferenceVariational Bayes Applications Solution Solve

  25. IntroductionVariationalInferenceVariational Bayes Applications Result: A Global Approximation Learning Rulesforexpectedrewardprobabilityandtheuncertaintyaboutit  Sequential Learning Algorithm True Posterior Laplace Variational Bayes

  26. IntroductionVariationalInferenceVariational Bayes Applications VB forBayesian Model Selection • Hence, ifis uniform . • Problem: • is “intractable” • Solution: • Justification: • If, then

  27. Motivation & OverviewVI IntuitionVB MathsApplications Summary ApproximateBayesianInference StructuralApproximations Variational Bayes (Ensemble Learning) MeanfieldParametricApprox. Learning Rules, Model Selection

More Related