Probabilistic Graphical Models CMSC 691 UMBC Two Problems for - PowerPoint PPT Presentation

Probabilistic Graphical Models CMSC 691 UMBC

Two Problems for Graphical Models 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 Finding the normalizer Computing the marginals

Two Problems for Graphical Models 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 Finding the normalizer Computing the marginals 𝑎 = ෍ ෑ 𝜔 𝑑 (𝑦 𝑑 ) 𝑦 𝑑

Two Problems for Graphical Models 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 Finding the normalizer Computing the marginals Sum over all variable combinations, with the x n coordinate fixed 𝑎 𝑜 (𝑤) = ෍ ෑ 𝜔 𝑑 (𝑦 𝑑 ) 𝑎 = ෍ ෑ 𝜔 𝑑 (𝑦 𝑑 ) 𝑦:𝑦 𝑜 =𝑤 𝑑 𝑦 𝑑 Example: 3 variables, fix the 2 nd dimension 𝑎 2 (𝑤) = ෍ ෍ ෑ 𝜔 𝑑 (𝑦 = 𝑦 1 , 𝑤, 𝑦 3 ) 𝑦 1 𝑦 3 𝑑

Two Problems for Graphical Models 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 Finding the normalizer Computing the marginals Sum over all variable combinations, with the x n coordinate fixed 𝑎 𝑜 (𝑤) = ෍ ෑ 𝜔 𝑑 (𝑦 𝑑 ) 𝑎 = ෍ ෑ 𝜔 𝑑 (𝑦 𝑑 ) 𝑦:𝑦 𝑜 =𝑤 𝑑 𝑦 𝑑 Example: 3 Q : Why are these difficult? variables, fix the 2 nd dimension A : Many different combinations 𝑎 2 (𝑤) = ෍ ෍ ෑ 𝜔 𝑑 (𝑦 = 𝑦 1 , 𝑤, 𝑦 3 ) 𝑦 1 𝑦 3 𝑑

Probabilistic Graphical Models A graph G that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂

Probabilistic Graphical Models A graph G that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Graph G = (vertices V, edges E) Distribution 𝑞(𝑌 1 , … , 𝑌 𝑂 )

Probabilistic Graphical Models A graph G that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Graph G = (vertices V, edges E) Distribution 𝑞(𝑌 1 , … , 𝑌 𝑂 ) Vertices ↔ random variables Edges show dependencies among random variables

Probabilistic Graphical Models A graph G that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Graph G = (vertices V, edges E) Distribution 𝑞(𝑌 1 , … , 𝑌 𝑂 ) Vertices ↔ random variables Edges show dependencies among random variables Two main flavors: directed graphical models and undirected graphical models

Outline Directed Graphical Models Undirected Graphical Models Factor Graphs

Directed Graphical Models A directed (acyclic) graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes into factors of 𝑌 𝑗 conditioned on the parents of 𝑌 𝑗

Directed Graphical Models A directed (acyclic) graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes into factors of 𝑌 𝑗 conditioned on the parents of 𝑌 𝑗 Benefit: read the independence properties are transparent

Directed Graphical Models A directed (acyclic) graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes into factors of 𝑌 𝑗 conditioned on the parents of 𝑌 𝑗 A graph/joint distribution that follows this is a Bayesian network

Bayesian Networks: Directed Acyclic Graphs 𝑦 1 𝑦 2 𝑦 3 5 𝑦 4 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = ෑ 𝑞 𝑦 𝑗 𝜌(𝑦 𝑗 )) 𝑗 “parents of” topological sort

Bayesian Networks: Directed Acyclic Graphs 𝑦 1 𝑦 2 𝑦 3 5 𝑦 4 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = ෑ 𝑞 𝑦 𝑗 𝜌(𝑦 𝑗 )) 𝑗 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , 𝑦 4 , 𝑦 5 = ???

Bayesian Networks: Directed Acyclic Graphs 𝑦 1 𝑦 2 𝑦 3 5 𝑦 4 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , 𝑦 4 , 𝑦 5 = 𝑞 𝑦 1 𝑞 𝑦 3 𝑞 𝑦 2 𝑦 1 , 𝑦 3 𝑞 𝑦 4 𝑦 2 , 𝑦 3 𝑞(𝑦 5 |𝑦 2 , 𝑦 4 )

Bayesian Networks: Directed Acyclic Graphs 𝑦 1 𝑦 2 𝑦 3 5 𝑦 4 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = ෑ 𝑞 𝑦 𝑗 𝜌(𝑦 𝑗 )) 𝑗 exact inference in general DAGs is NP-hard inference in trees can be exact

Directed Graphical Model Notation 𝑦 1 𝑦 2 𝑦 3 5 𝑦 4 Unshaded nodes Shaded nodes are are unobserved observed R.V.s (latent) R.V.s

D-Separation: Testing for Conditional Independence d-separation X & Y are d-separated if for all paths P, one of the following is true: Variables X & Y are P has a chain with an observed middle node conditionally independent given Z if all X Y (undirected) paths from P has a fork with an observed parent node (any variable in) X to (any variable in) Y are X Y d-separated by Z P includes a “v - structure” or “collider” with all unobserved descendants X Z Y

D-Separation: Testing for Conditional Independence d-separation Variables X & Y are conditionally independent given Z if all (undirected) paths from (any variable X & Y are d-separated if for all paths P, one of in) X to (any variable in) Y are d-separated by Z the following is true: P has a chain with an observed middle node observing Z blocks the path from X to Y X Z Y P has a fork with an observed parent node Z observing Z blocks the path from X to Y X Y P includes a “v - structure” or “collider” with all unobserved descendants X Z Y not observing Z blocks the path from X to Y

D-Separation: Testing for Conditional Independence d-separation Variables X & Y are conditionally independent given Z if all (undirected) paths from (any variable X & Y are d-separated if for all paths P, one of in) X to (any variable in) Y are d-separated by Z the following is true: P has a chain with an observed middle node observing Z blocks the path from X to Y X Z Y P has a fork with an observed parent node Z observing Z blocks the path from X to Y X Y P includes a “v - structure” or “collider” with all unobserved descendants not observing Z blocks X Z Y the path from X to Y 𝑞 𝑦, 𝑧, 𝑨 = 𝑞 𝑦 𝑞 𝑧 𝑞(𝑨|𝑦, 𝑧) 𝑞 𝑦, 𝑧 = ෍ 𝑞 𝑦 𝑞 𝑧 𝑞(𝑨|𝑦, 𝑧) = 𝑞 𝑦 𝑞 𝑧 𝑨

Markov Blanket the set of nodes needed to form the complete conditional for a variable x i 𝑞(𝑦 1 , … , 𝑦 𝑂 ) 𝑞 𝑦 𝑗 𝑦 𝑘≠𝑗 = ∫ 𝑞 𝑦 1 , … , 𝑦 𝑂 𝑒𝑦 𝑗 x ς 𝑙 𝑞(𝑦 𝑙 |𝜌 𝑦 𝑙 ) factorization = of graph ∫ ς 𝑙 𝑞 𝑦 𝑙 𝜌 𝑦 𝑙 ) 𝑒𝑦 𝑗 factor out terms not dependent on x i Markov blanket of a node x ς 𝑙:𝑙=𝑗 or 𝑗∈𝜌 𝑦 𝑙 𝑞(𝑦 𝑙 |𝜌 𝑦 𝑙 ) is its parents, children, and = children's parents ∫ ς 𝑙:𝑙=𝑗 or 𝑗∈𝜌 𝑦 𝑙 𝑞 𝑦 𝑙 𝜌 𝑦 𝑙 ) 𝑒𝑦 𝑗 (in this example, shading does not show observed/latent)

Outline Directed Graphical Models Undirected Graphical Models Factor Graphs

Undirected Graphical Models An undirected graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes based on cliques in the graph

Undirected Graphical Models An undirected graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes based on cliques in the graph Common name: Markov Random Fields

Undirected Graphical Models An undirected graph G=(V,E) that represents a probability distribution over random variables 𝑌 1 , … , 𝑌 𝑂 Joint probability factorizes based on cliques in the graph Common name: Markov Random Fields Undirected graphs can have an alternative formulation as Factor Graphs

Markov Random Fields: Undirected Graphs 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂

Markov Random Fields: Undirected Graphs clique : subset of nodes, where nodes are pairwise connected maximal clique : a clique that cannot add a node and remain a clique 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂

Markov Random Fields: Undirected Graphs clique : subset of nodes, where nodes are pairwise connected maximal clique : a clique that cannot add a node and remain a clique 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 variables part of the clique C global normalization maximal potential function (not cliques necessarily a probability!)

Markov Random Fields: Undirected Graphs clique : subset of nodes, where nodes are pairwise connected maximal clique : a clique that cannot add a node and remain a clique 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 variables part of the clique C global normalization maximal potential function (not cliques necessarily a probability!)

Markov Random Fields: Undirected Graphs clique : subset of nodes, where nodes are pairwise connected maximal clique : a clique that cannot add a node and remain a clique 𝑞 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 𝑂 = 1 𝑎 ෑ 𝜔 𝐷 𝑦 𝑑 𝐷 variables part Q : What restrictions should we of the clique C place on the potentials 𝜔 𝐷 ? global normalization maximal potential function (not cliques necessarily a probability!)