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 LinearBranchingApplication DirectedMarkov Chain (HMM) Bayesian network (BN) AI Statistics UndirectedLinear chain conditional random field (CRF) Markov network (MN) Physics (Ising) Image/Vision Both directed and undirected arcs: chain graphs

Bayesian Network Structure Directed acyclic graph G –Nodes X 1,…,X n represent random variables G encodes local Markov assumptions –X i is independent of its non-descendants given its parents A BC E G DF

A simple example We want to model whether the grass is wet The grass can get wet if –It rains –If the sprinklers are on Whether it rains depends on whether it is cloudy When it is cloudy we are less likely to turn on the sprinklers

A Simple Example Variables –Cloudy (C), Rain (R), Sprinkler (S), GrassWet (W) CRSWProb. FFFF0.01 FFFT0.04 FFTF0.05 FFTT0.01 FTFF0.02 FTFT0.07 FTTF0.2 FTTT0.1 TFFF0.01 TFFT0.07 TFTF0.13 TFTT0.04 TTFF0.06 TTFT0.05 TTTF0.1 TTTT independent parameters

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 TATTAA TTAAAA TAGAAA T C A G 11 A1A1 22 A2A2 33 A3A3 44 A4A4 55 A5A5 66 A6A6 The nucleotide distributions at different sites are independent !

Mixture of profile model A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 Z 1111 m1m1 1212 m2m2 1414 m4m4 1515 m5m5 The nt-distributions at different sites are conditionally independent but marginally dependent !

Tree model 11 A1A1 22 A2A2 33 A3A3 44 A4A4 55 A5A5 66 A6A6 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 –Do not have to sum to 1 –Represent affinities D A BC AC  1 [A,C] a0a0 c0c0 4 a0a0 c1c1 12 a1a1 c0c0 2 a1a1 c1c1 9 AB  2 [A,B] a0a0 b0b0 30 a0a0 b1b1 5 a1a1 b0b0 1 a1a1 b1b1 10 CD  3 [C,D] c0c0 d0d0 30 c0c0 d1d1 5 c1c1 d0d0 1 c1c1 d1d1 10 BD  4 [B,D] b0b0 d0d0 100 b0b0 d1d1 1 b1b1 d0d0 1 b1b1 d1d1 1000

Markov network Represents joint distribution –Unnormalized factor –Partition function –Probability D A BC

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 C A DB C A DB Maximal cliques {A,B} {B,C} {C,D} {A,D} 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 {  [X i ]:i=1,...n} –A set of edge potentials {  [X i,X j ]: X i,X j  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 X i corresponds to pixel i –Domain is {1,...K} –Value represents region assignment to pixel i Neighboring pixels are connected in the network Appearance distribution –w i k – extent to which pixel i “fits” region k (e.g., difference from typical pixel for region k) –Introduce node potential exp(-w i k 1{X i =k}) Edge potentials –Encodes contiguity preference by edge potential exp( 1{X i =X j }) for >0

Markov Network for Segmentation Solution: inference –Find most likely assignment to X i variables X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 Appearance distribution Contiguity preference