Lecture 9: Undirected Graphical Models Machine Learning
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1 Lecture 9: Undirected Graphical Models Machine Learning Andrew Rosenberg March 5, /1
2 Today Graphical Models Probabilities in Undirected Graphs 2/1
3 Undirected Graphs What if we allow undirected graphs? What do they correspond to? It s not cause/effect, or trigger/response, rather, general dependence. Example: Image pixels, where each pixel is a bernouli. Can have a probability over all pixels p(x 11,x 1M,x M1,x MM ) Bright pixels have bright neighbors. No parents, just probabilities. Grid models are called Markov Random Fields. 3/1
4 Undirected Graphs x x y {w} cannot represent x y {w,z} Undirected separation is easy. z w y x z w y x y {w,z} w z {x,y} To check x a x b x c, check Graph reachability of x a and x b without going through nodes in x c. 4/1
5 Undirected Graphs x y x y x y z z z x y x y z OR x z x y z Undirected separation is easy. To check x a x b x c, check Graph reachability of x a and x b without going through nodes in x c. 5/1
6 Probabilities in Undirected Graphs Graph cliques define clusters of dependent variables d f a b c e Clique: a set of nodes such there is an edge between every pair of members of the set. We define probability as a product of functions defined over cliques 6/1
7 Probabilities in Undirected Graphs Graph cliques define clusters of dependent variables d a b c f e Clique: a set of nodes such there is an edge between every pair of members of the set. We define probability as a product of functions defined over cliques 7/1
8 Probabilities in Undirected Graphs Graph cliques define clusters of dependent variables d a b c f e Clique: a set of nodes such there is an edge between every pair of members of the set. We define probability as a product of functions defined over cliques 8/1
9 Probabilities in Undirected Graphs Graph cliques define clusters of dependent variables d f a b c e Clique: a set of nodes such there is an edge between every pair of members of the set. We define probability as a product of functions defined over cliques 9/1
10 Probabilities in Undirected Graphs Graph cliques define clusters of dependent variables d f a b c e Clique: a set of nodes such there is an edge between every pair of members of the set. We define probability as a product of functions defined over cliques 10 /1
11 Representing Probabilities Potential Functions over cliques p(x) = p(x 0,...,x n 1 ) = 1 Z ψ c (x c ) c C Normalizing Term guarantees that p(x) sums to 1 Z = ψ c (x c ) x c C Potential Functions are positive functions over groups of connected variables. Use only maximal cliques. e.g. ψ(x 1,x 2,x 3 )ψ(x 2,x 3 ) ψ(x 1,x 2,x 3 ) 11 /1
12 Logical Inference a NOT b e XOR c AND d In Logic Networks, nodes are binary, and edges represent gates Gates: AND, OR, XOR, NAND, NOR, NOT, etc. Inference: given observed variables, predict others. Problems: Uncertainty, conflicts and inconsistency Rather than saying a variable is True or False, let s say it is.8 True and.2 False. Probabilistic Inference 12 /1
13 Inference Probabilistic Inference a NOT b e XOR c AND d Replace the logic network with a Bayesian Network Probabilistic Inference: given observed variables, predict marginals over others. Not b=t b=f a=t 0 1 a=f /1
14 Inference Probabilistic Inference a NOT b e XOR c AND d Replace the logic network with a Bayesian Network Probabilistic Inference: given observed variables, predict marginals over others. Soft Not b=t b=f a=t.1.9 a=f /1
15 Inference General Problem Given a graph and probabilities, for any subset of variables, find p(x e x o ) = p(x e,x o ) p(x o ) Compute both marginals and divide. But this can be exponential...(based on the number of parent each node has, or the size of the cliques) p(x j,x k ) = x 0 p(x j,x k ) = x 0 M 1... p(x i π i ) x 1 x M 1 i=0... ψ(x c ) x 1 x M 1 c C Have efficient learning and storage in Graphical Models, now inference. 15 /1
16 Inefficient Marginals Brute Force. Given CPTs and a graph structure we can compute arbitrary marginals by brute force, but it s inefficient. For Example p(x) = p(x 0)p(x 1 x 0)p(x 2 x 0)p(x 3 x 1)p(x 4 x 2)p(x 5 x 2,x 5) p(x 0, x 2) = p(x 0)p(x 2 x 0) X p(x 0, x 5) = p(x 0)p(x 1 x 0)p(x 2 x 0)p(x 3 x 1)p(x 4 x 2)p(x 5 x 2,x 5) x 1,x 2,x 3,x 4 P x p(x 0 x 5) = 1,x 2,x 3,x 4 p(x) P x 0,x 1,x 2,x 3,x 4 p(x) P x p(x 0 x 5 = TRUE) = 1,x 2,x 3,x 4 p(x U\5 x 5 = TRUE) P x 0,x 1,x 2,x 3,x 4 p(x U\5 x 5 = TRUE) 16 /1
17 Efficient Computation of Marginals b a c e d Pass messages (small tables) around the graph. The messages will be small functions that propagate potentials around an undirected graphical model. The inference technique is the Junction Tree Algorithm 17 /1
18 Junction Tree Algorithm Efficient Message Passing on Undirected Graphs. For Directed Graphs, first convert to an Undirected Graph (Moralization). Moralization Introduce Evidence Triangulate Construct Junction Tree Propagate Probabilities Junction Tree Algorithm 18 /1
19 Bye Next Junction Tree Algorithm 19 /1
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