Probabilistic graphical models Probabilistic graphical models Graphical models
Probabilistic graphical models
Probabilistic graphical models • Graphical models are a marriage between probability theory and graph theory (Michael Jordan, 1998) • A compact representation of joint probability distributions. • Graphs – nodes: random variables (probabilistic distribution over a fixed alphabet) – edges (arcs), or lack of edges: conditional independence assumptions
Classification of probabilistic graphical models Linear Directed Markov Chain (HMM) Undirected Linear chain conditional random field (CRF) Branching Application Bayesian network (BN) AI Statistics Markov network (MN) Physics (Ising) Image/Vision Both directed and undirected arcs: chain graphs
Bayesian Network Structure • Directed acyclic graph G – Nodes X 1, …, Xn represent random variables • G encodes local Markov assumptions – Xi is independent of its non-descendants given its parents A D B C E F G
Bayesian Network • Conditional probability distribution (CPD) at each node – T (true), F (false) • P(C, S, R, W) = P(C) * P(S|C) * P(R|C, S) * P(W|C, S, R) P(C) * P(S|C) * P(R|C) * P(W|S, R) • 8 independent parameters
Training Bayesian network: frequencies Known: frequencies Pr(c, s, r, w) for all (c, s, r, w)
Application: Recommendation Systems • Given user preferences, suggest recommendations – Amazon. com • Input: movie preferences of many users • Solution: model correlations between movie features – Users that like comedy, often like drama – Users that like action, often do not like cartoons – Users that like Robert Deniro films often like Al Pacino films – Given user preferences, can predict probability that new movies match preferences
Application: modeling DNA motifs • Profile model: no dependences between positions • Markov model: dependence between adjacent positions • Bayesian network model: non-local dependences
A DNA profile TATAAA TATAAT TATAAA TATTAAAA TAGAAA 1 8 0 0 0 T C A G 1 2 1 0 7 0 3 2 A 1 3 6 0 1 1 A 2 4 1 0 7 0 4 A 3 5 0 0 8 0 5 A 4 6 1 0 7 0 6 A 5 The nucleotide distributions at different sites are independent ! A 6
Mixture of profile model 11 m 1 12 A 1 m 2 A 2 14 A 3 m 4 A 4 15 A 5 Z The nt-distributions at different sites are conditionally independent but marginally dependent ! m 5 A 6
Tree model 1 3 2 A 1 A 2 4 A 3 5 A 4 6 A 5 A 6 The nt-distributions at different sites are pairwisely dependent !
Undirected graphical models (e. g. Markov network) • Useful when edge directionality cannot be assigned • Simpler interpretation of structure – Simpler inference – Simpler independency structure • Harder to learn
Markov network • Nodes correspond to random variables • Local factor models are attached to sets of nodes – Factor elements are positive A – Do not have to sum to 1 A C 1[A, C] a 0 c 0 4 a 0 – Represent affinities 0 1 0 B 2[A, B] b 0 30 a c 12 a b 1 5 a 1 c 0 2 a 1 b 0 1 a 1 c 1 9 a 1 b 1 10 A C C D 3[C, D] c 0 d 0 30 c 0 d 1 c 1 B B D 4[B, D] b 0 d 0 100 5 b 0 d 1 1 d 0 1 b 1 d 0 1 d 1 10 b 1 d 1 1000 D
Markov network • Represents joint distribution – Unnormalized factor A – Partition function C – Probability B D
Markov Network Factors • A factor is a function from value assignments of a set of random variables D to real positive numbers – The set of variables D is the scope of the factor • Factors generalize the notion of CPDs – Every CPD is a factor (with additional constraints)
Markov Network Factors A A B D C Maximal cliques • {A, B} • {B, C} • {C, D} • {A, D} B D C Maximal cliques • {A, B, C} • {A, C, D}
Pairwise Markov networks • A pairwise Markov network over a graph H has: – A set of node potentials { [Xi]: i=1, . . . n} – A set of edge potentials { [Xi, Xj]: Xi, Xj H} – Example: Grid structured Markov network X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34
Application: Image analysis • The image segmentation problem – Task: Partition an image into distinct parts of the scene – Example: separate water, sky, background
Markov Network for Segmentation • Grid structured Markov network • Random variable Xi corresponds to pixel i – Domain is {1, . . . K} – Value represents region assignment to pixel i • Neighboring pixels are connected in the network • Appearance distribution – wik – extent to which pixel i “fits” region k (e. g. , difference from typical pixel for region k) – Introduce node potential exp(-wik 1{Xi=k}) • Edge potentials – Encodes contiguity preference by edge potential exp( 1{Xi=Xj}) for >0
Markov Network for Segmentation Appearance distribution X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 • Solution: inference Contiguity preference – Find most likely assignment to Xi variables
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