ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning

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1 ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning Topics Markov Random Fields: Inference Exact: VE Exact+Approximate: BP Readings: Barber 5 Dhruv Batra Virginia Tech

2 Administrativia HW3 Out 2 days ago Due: Apr 4, 11:55pm Implementation: Loopy Belief Propagation in MRFs (C) Dhruv Batra 2

3 Recap of Last Time (C) Dhruv Batra 3

4 Markov Nets Set of random variables Undirected graph Encodes independence assumptions Unnormalized Factor Tables Joint distribution: Product of Factors (C) Dhruv Batra 4

5 Pairwise MRFs Pairwise Factors A function of 2 variables Often unary terms are also allowed (although strictly speaking unnecessary) On board (C) Dhruv Batra 5

6 Pairwise MRF: Example (C) Dhruv Batra 6

7 Normalization for computing probabilities To compute actual probabilities, must compute normalization constant (also called partition function) Computing partition function is hard! Must sum over all possible assignments (C) Dhruv Batra Slide Credit: Carlos Guestrin 7

8 Nearest-Neighbor Grids Low Level Vision Image denoising Stereo Optical flow Shape from shading Superresolution Segmentation unobserved or hidden variable local observation (C) Dhruv Batra Slide Credit: Erik Sudderth 8

9 Factorization in Markov networks Given an undirected graph H over variables X={X 1,...,X n } A distribution P factorizes over H if there exist subsets of variables D 1 X,, D m X, such that D i are fully connected in H non-negative potentials (or factors) φ 1 (D 1 ),, φ m (D m ) also known as clique potentials such that Also called Markov random field H, or Gibbs distribution over H (C) Dhruv Batra Slide Credit: Carlos Guestrin 9

10 Structure in cliques Possible potentials for this graph: A B C

11 Factor graphs Bipartite graph: variable nodes (ovals) for X 1,,X n factor nodes (squares) for φ 1,,φ m edge X i φ j if X i ε Scope[φ j ] A C B Very useful for approximate inference Make factor dependency explicit

12 Types of Graphical Models Directed Factor Undirected (C) Dhruv Batra Slide Credit: Erik Sudderth 12

13 Plan for today MRF Inference Exact Inference Variable Elimination Exact+Approximate Inference (General) Belief Propagation Cluster Graphs Family Preserving Property Running Intersection Property Message-Passing Approximate Inference Bethe Cluster Graph Loopy BP Exact Inference Junction Tree BP on Junction Trees (C) Dhruv Batra 13

14 Marginal Inference Example Evidence: E=e (e.g. N=t) Query variables of interest Y Flu Sinus Allergy Headache Nose=t Conditional Probability: P(Y E=e) P(F N=t) (C) Dhruv Batra 14

15 Variable Elimination algorithm Given a BN and a query P(Y e) P(Y,e) Instantiate Evidence IMPORTANT!!! Choose an ordering on variables, e.g., X 1,, X n For i = 1 to n, If X i {Y,E} Collect factors f 1,,f k that include X i Generate a new factor by eliminating X i from these factors Variable X i has been eliminated! Normalize P(Y,e) to obtain P(Y e) (C) Dhruv Batra Slide Credit: Carlos Guestrin 15

16 VE for MRF Exactly the same algorithm works! Factors are no longer CPTs But VE doesn t care (C) Dhruv Batra 16

17 Example Chain MRF Compute: X 1 X 2 X 3 X 4 X 5 P (X 1 X 5 = x 5 ) VE steps on board (C) Dhruv Batra 17

18 Example Chain MRF Compute: X 1 X 2 X 3 X 4 X 5 P (X i X 5 = x 5 ) i {1, 2, 3, 4} Variable elimination for every i, what s the complexity? Can we do better by caching intermediate results? Yes! via Junction-Trees But let s look at BP first (C) Dhruv Batra 18

19 New Topic: Belief Propagation (C) Dhruv Batra Image Credit: LAMatters 19

20 What is BP? Technique invented by Judea Pearl in 1982 Initially to compute marginals in BNs Later generalized to MRFs, Factor Graphs To MAP inference; to Marginal-MAP inference Lots of analysis Under some cases EXACT Tree graphs In this setting, BP equivalent to VE on Junction-Trees Submodular potentials In this setting, BP equivalent to Graph-Cuts! [Tarlow et al. UAI11] In general Approximate (C) Dhruv Batra 20

21 Message Passing Variables/Factors talk to each other via messages: I (variable X 3 ) think that you (variable X 2 ): belong to state 1 with confidence 0.4 belong to state 2 with confidence 10 belong to state 3 with confidence 1.5 (C) Dhruv Batra Image Credit: James Coughlan 21

22 Overview of BP Pick a graph to pass messages on Cluster Graph Pick an ordering of edges Round-robin Leaves-Root-Leaves on a tree Asynchonous Till convergence or exhaustion: Pass messages on edges At vertices on graph compute psuedo-marginals (C) Dhruv Batra 22

23 Cluster graph Cluster Graph: For set of factors F Undirected graph Each node i associated with a cluster C i Each edge i j is associated with a separator set of variables S ij C i C j (C) Dhruv Batra Slide Credit: Carlos Guestrin 23

24 Generalized BP Initialization: Assign each factor φ to a cluster α(φ), Scope[φ] C α(φ) Initialize cluster: Initialize messages: While not converged, send messages: Belief: On Board (C) Dhruv Batra Slide Credit: Carlos Guestrin 24

25 Properties of Cluster Graphs Family preserving: For set of factors F for each factor f! j F,! node i such that scope[f i ] C i (C) Dhruv Batra Slide Credit: Carlos Guestrin 25

26 Properties of Cluster Graphs Running intersection property (RIP) If X! C i and X! C j then! one and only one path from C i to C j where X! S uv for every edge (u,v) in the path (C) Dhruv Batra Slide Credit: Carlos Guestrin 26

27 Two cluster graph satisfying RIP with different edge sets (C) Dhruv Batra Slide Credit: Carlos Guestrin 27

28 Overview of BP Pick a graph to pass messages on Cluster Graph Pick an ordering of edges Round-robin Leaves-Root-Leaves on a tree Asynchonous Till convergence or exhaustion: Pass messages on edges At vertices on graph compute psuedo-marginals (C) Dhruv Batra 28

29 Cluster Graph for Loopy BP Bethe Cluster Graph Set of Clusters = Factors F! {X i } Sometimes also called Running BP on Factor Graphs Example on board Does the Bethe Cluster Graph satisfy properties? (C) Dhruv Batra 29

30 Loopy BP in Factor graphs From node i to factor j: F(i) factors whose scope includes X i A B C D E ABC ABD BDE CDE From factor j to node i: Scope[φ j ] = Y! {X i } Belief: Node: Factor: (C) Dhruv Batra Slide Credit: Carlos Guestrin 30

31 Loopy BP on Pairwise Markov Nets δ i j (y j )= φ i (y i )φ ij (y i,y j ) δ k i (y i ) y i k N (i) j y i y j x j x i (C) Dhruv Batra 31