Graphical Models Factor Graphs Test-time Inference Training Part 2: Introduction to Graphical Models Sebastian Nowozin and Christoph H. Lampert Colorado Springs, 25th June 2011 Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Introduction ◮ Model: relating observations x to quantities of interest y f ◮ Example 1: given RGB image x , infer depth y for each pixel X Y ◮ Example 2: given RGB image x , infer presence and positions y of all objects f : X → Y shown Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Introduction ◮ Model: relating observations x to quantities of interest y f ◮ Example 1: given RGB image x , infer depth y for each pixel X Y ◮ Example 2: given RGB image x , infer presence and positions y of all objects f : X → Y shown X : image, Y : object annotations Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Introduction ◮ General case: mapping x ∈ X to y ∈ Y ◮ Graphical models are a concise x language to define this mapping f ( x ) ◮ Mapping can be ambiguous : X Y measurement noise, lack of well-posedness (e.g. occlusions) f : X → Y ◮ Probabilistic graphical models: define form p ( y | x ) or p ( x , y ) for all y ∈ Y Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Introduction ◮ General case: mapping x ∈ X to y ∈ Y ? ◮ Graphical models are a concise language to define this mapping x ◮ Mapping can be ambiguous : ? X Y measurement noise, lack of well-posedness (e.g. occlusions) p ( Y | X = x ) ◮ Probabilistic graphical models: define form p ( y | x ) or p ( x , y ) for all y ∈ Y Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Graphical Models A graphical model defines ◮ a family of probability distributions over a set of random variables, ◮ by means of a graph, ◮ so that the random variables satisfy conditional independence assumptions encoded in the graph. Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Graphical Models A graphical model defines ◮ a family of probability distributions over a set of random variables, ◮ by means of a graph, ◮ so that the random variables satisfy conditional independence assumptions encoded in the graph. Popular classes of graphical models, ◮ Undirected graphical models (Markov random fields), ◮ Directed graphical models (Bayesian networks), ◮ Factor graphs, ◮ Others: chain graphs, influence diagrams, etc. Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Bayesian Networks ◮ Graph: G = ( V , E ), E ⊂ V × V Y i Y j ◮ directed ◮ acyclic ◮ Variable domains Y i Y k ◮ Factorization � p ( Y = y ) = p ( y i | y pa G ( i ) ) Y l i ∈ V A simple Bayes net over distributions, by conditioning on parent nodes. ◮ Example p ( Y = y ) = p ( Y l = y l | Y k = y k ) p ( Y k = y k | Y i = y i , Y j = y j ) p ( Y i = y i ) p ( Y j = y j ) . Sebastian Nowozin and Christoph H. Lampert ◮ Family of distributions Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Bayesian Networks ◮ Graph: G = ( V , E ), E ⊂ V × V Y i Y j ◮ directed ◮ acyclic ◮ Variable domains Y i Y k ◮ Factorization � p ( Y = y ) = p ( y i | y pa G ( i ) ) Y l i ∈ V A simple Bayes net over distributions, by conditioning on parent nodes. ◮ Example p ( Y = y ) = p ( Y l = y l | Y k = y k ) p ( Y k = y k | Y i = y i , Y j = y j ) p ( Y i = y i ) p ( Y j = y j ) . Sebastian Nowozin and Christoph H. Lampert ◮ Family of distributions Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Undirected Graphical Models Y i Y j Y k ◮ = Markov random field (MRF) = Markov network A simple MRF ◮ Graph: G = ( V , E ), E ⊂ V × V ◮ undirected, no self-edges ◮ Variable domains Y i ◮ Factorization over potentials ψ at cliques , p ( y ) = 1 � ψ C ( y C ) Z C ∈C ( G ) ◮ Constant Z = � � C ∈C ( G ) ψ C ( y C ) y ∈Y ◮ Example p ( y ) = 1 Z ψ i ( y i ) ψ j ( y j ) ψ l ( y l ) ψ i , j ( y i , y j ) Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Undirected Graphical Models Y i Y j Y k ◮ = Markov random field (MRF) = Markov network A simple MRF ◮ Graph: G = ( V , E ), E ⊂ V × V ◮ undirected, no self-edges ◮ Variable domains Y i ◮ Factorization over potentials ψ at cliques , p ( y ) = 1 � ψ C ( y C ) Z C ∈C ( G ) ◮ Constant Z = � � C ∈C ( G ) ψ C ( y C ) y ∈Y ◮ Example p ( y ) = 1 Z ψ i ( y i ) ψ j ( y j ) ψ l ( y l ) ψ i , j ( y i , y j ) Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Example 1 Y i Y j Y k ◮ Cliques C ( G ): set of vertex sets V ′ with V ′ ⊆ V , E ∩ ( V ′ × V ′ ) = V ′ × V ′ ◮ Here C ( G ) = {{ i } , { i , j } , { j } , { j , k } , { k }} ◮ p ( y ) = 1 Z ψ i ( y i ) ψ j ( y j ) ψ l ( y l ) ψ i , j ( y i , y j ) Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Graphical Models Example 2 Y i Y j Y k Y l ◮ Here C ( G ) = 2 V : all subsets of V are cliques ◮ p ( y ) = 1 � ψ A ( y A ) . Z A ∈ 2 { i , j , k , l } Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Factor Graphs ◮ Graph: G = ( V , F , E ), E ⊆ V × F Y i Y j ◮ variable nodes V , ◮ factor nodes F , ◮ edges E between variable and factor nodes. ◮ scope of a factor, N ( F ) = { i ∈ V : ( i , F ) ∈ E} Y k Y l ◮ Variable domains Y i ◮ Factorization over potentials ψ at factors , Factor graph p ( y ) = 1 � ψ F ( y N ( F ) ) Z F ∈F ◮ Constant Z = � � F ∈F ψ F ( y N ( F ) ) y ∈Y Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Factor Graphs ◮ Graph: G = ( V , F , E ), E ⊆ V × F Y i Y j ◮ variable nodes V , ◮ factor nodes F , ◮ edges E between variable and factor nodes. ◮ scope of a factor, N ( F ) = { i ∈ V : ( i , F ) ∈ E} Y k Y l ◮ Variable domains Y i ◮ Factorization over potentials ψ at factors , Factor graph p ( y ) = 1 � ψ F ( y N ( F ) ) Z F ∈F ◮ Constant Z = � � F ∈F ψ F ( y N ( F ) ) y ∈Y Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Why factor graphs? Y i Y j Y i Y j Y i Y j Y k Y l Y k Y l Y k Y l ◮ Factor graphs are explicit about the factorization ◮ Hence, easier to work with ◮ Universal (just like MRFs and Bayesian networks) Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Capacity Y i Y j Y i Y j Y k Y l Y k Y l ◮ Factor graph defines family of distributions ◮ Some families are larger than others Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Four remaining pieces 1. Conditional distributions (CRFs) 2. Parameterization 3. Test-time inference 4. Learning the model from training data Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Four remaining pieces 1. Conditional distributions (CRFs) 2. Parameterization 3. Test-time inference 4. Learning the model from training data Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
Graphical Models Factor Graphs Test-time Inference Training Factor Graphs Conditional Distributions X i X j ◮ We have discussed p ( y ), ◮ How do we define p ( y | x )? ◮ Potentials become a function of x N ( F ) ◮ Partition function depends on x Y i Y j ◮ Conditional random fields (CRFs) ◮ x is not part of the probability model, i.e. not conditional treated as random variable distribution Sebastian Nowozin and Christoph H. Lampert Part 2: Introduction to Graphical Models
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