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Bishop's tutorial on graphical models

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					Machine Learning Techniques
   for Computer Vision
     Part 1: Graphical Models

       Christopher M. Bishop
       Microsoft Research Cambridge




           ECCV 2004, Prague
About this Tutorial
• Learning is the new frontier in computer vision
• Focus on concepts
   – not lists of algorithms
   – not technical details
• Graduate level
• Please ask questions!




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Overview
• Part 1: Graphical models
   – directed and undirected graphs
   – inference and learning
• Part 2: Unsupervised learning
   – mixture models, EM
   – variational inference, model complexity
   – continuous latent variables
• Part 3: Supervised learning
   – decision theory
   – linear models, neural networks,
   – boosting, sparse kernel machines


Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Probability Theory
• Sum rule



• Product rule



• From these we have Bayes’ theorem



     – with normalization



Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Role of the Graphs
• New insights into existing models
• Motivation for new models
• Graph based algorithms for calculation and computation
   – c.f. Feynman diagrams in physics




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Decomposition
• Consider an arbitrary joint distribution


• By successive application of the product rule




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Directed Acyclic Graphs
• Joint distribution



    where           denotes the parents of i




                              No directed cycles


Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Undirected Graphs
• Provided           then joint distribution is product of
  non-negative functions over the cliques of the graph




    where          are the clique potentials, and Z is a
    normalization constant




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Conditioning on Evidence
• Variables may be hidden (latent) or visible (observed)




• Latent variables may have a specific interpretation, or
  may be introduced to permit a richer class of distribution
Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Conditional Independences
• x independent of y given z if, for all values of z,




• For undirected graphs this is given by graph separation!




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
“Explaining Away”
• C.I. for directed graphs similar, but with one subtlety
• Illustration: pixel colour in an image




                    lighting                              surface
                     colour                                colour



                                     image colour




Machine Learning Techniques for Computer Vision (ECCV 2004)         Christopher M. Bishop
Directed versus Undirected




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Example: State Space Models
• Hidden Markov model
• Kalman filter




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Example: Bayesian SSM




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Example: Factorial SSM
• Multiple hidden sequences
• Avoid exponentially large hidden space




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Example: Markov Random Field
• Typical application: image region labelling




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Example: Conditional Random Field




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Inference
• Simple example: Bayes’ theorem




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Message Passing
• Example




• Find marginal for a particular node




     – for M-state nodes, cost is
     – exponential in length of chain
     – but, we can exploit the graphical structure
       (conditional independences)

Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Message Passing
• Joint distribution


• Exchange sums and products




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Message Passing
• Express as product of messages




• Recursive evaluation of messages




• Find Z by normalizing

Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Belief Propagation
• Extension to general tree-structured graphs
• At each node:
   – form product of incoming messages and local evidence
   – marginalize to give outgoing message
   – one message in each direction across every link




• Fails if there are loops
Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Junction Tree Algorithm
• An efficient exact algorithm for a general graph
   – applies to both directed and undirected graphs
   – compile original graph into a tree of cliques
   – then perform message passing on this tree
• Problem:
   – cost is exponential in size of largest clique
   – many vision models have intractably large cliques




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Loopy Belief Propagation
• Apply belief propagation directly to general graph
   – need to keep iterating
   – might not converge
• State-of-the-art performance in error-correcting codes




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Max-product Algorithm
• Goal: find


     – define


     – then



• Message passing algorithm with “sum” replaced by “max”
• Example:
   – Viterbi algorithm for HMMs

Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Inference and Learning
• Data set


• Likelihood function (independent observations)




• Maximize (log) likelihood


• Predictive distribution



Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Regularized Maximum Likelihood
• Prior             , posterior


• MAP (maximum posterior)


• Predictive distribution


• Not really Bayesian




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Bayesian Learning
• Key idea is to marginalize over unknown parameters,
  rather than make point estimates


   – avoids severe over-fitting of ML and MAP
   – allows direct model comparison
• Parameters are now latent variables
• Bayesian learning is an inference problem!




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Bayesian Learning




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Bayesian Learning




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
And Finally … the Exponential Family
• Many distributions can be written in the form


• Includes:
   – Gaussian
   – Dirichlet
   – Gamma
   – Multi-nomial
   – Wishart
   – Bernoulli
   –…
• Building blocks in graphs to give rich probabilistic models

Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Illustration: the Gaussian
• Use precision (inverse variance)




• In standard form




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Maximum Likelihood
• Likelihood function (independent observations)




• Depends on data via sufficient statistics of fixed dimension




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Conjugate Priors
• Prior has same functional form as likelihood



• Hence posterior is of the form




• Can interpret prior as effective observations of value
• Examples:
   – Gaussian for the mean of a Gaussian
   – Gaussian-Wishart for mean and precision of Gaussian
   – Dirichlet for the parameters of a discrete distribution
Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop
Summary of Part 1
• Directed graphs



• Undirected graphs



• Inference by message passing: belief propagation




Machine Learning Techniques for Computer Vision (ECCV 2004)   Christopher M. Bishop

				
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