Part I: Sum Product Algorithm and (Loopy) Belief Propagation. What s wrong with VarElim. Forwards algorithm (filtering) Forwards-backwards algorithm

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1 OU 56 Probabilistic Graphical Models Loopy Belief Propagation and lique Trees / Join Trees lides from Kevin Murphy s Graphical Model Tutorial (with minor changes) eading: Koller and Friedman h 0 Part I: um Product Algorithm and (Loopy) Belief Propagation (All you need to know is slide 7 which is covered in the excerpt from MacKay s book posted on the course webpage.) hat s wrong with VarElim Often we want to query all hidden nodes. VarElim takes O(N K w+ ) time to compute P(X i x e ) for all (hidden) nodes i. There exist message passing algorithms that can do this in O(N K w+ ) time. Later, we will use these to do approximate inference in O(N K ) time, indep of w. epeated variable elimination leads to redundant calculations X X X Y Y Y X X X Y Y Y P- O(N K ) time to compute all N marginals P- Forwards-backwards algorithm abiner89,etc Forwards algorithm (filtering) X t X t X t+ X t X t Y :t- +:N Y :t- Forwards prediction Local evidence Backwards prediction (Use dynamic programming to compute these) P-5 P-6

2 Backwards algorithm Forwards-backwards algorithm X t X t X t+ X t+ X X X + +:N b Forwards b b Backwards Backwards messages independent of forwards messages ombine P-7 O(N K ) time to compute all N marginals, not O(N K ) P-8 Belief propagation Pearl88,hafer90,Yedidia0,etc Absorbing messages Forwards-backwards algorithm can be generalized to apply to any tree-like graph (ones with no loops). For now, we assume pairwise potentials. X t- X t X t+ P-9 P-0 ending messages entralized protocol ollect to root (post-order) Distribute from root (pre-order) 5 5 X t- X t X t+ omputes all N marginals in passes over graph P- P-

3 Distributed protocol Loopy belief propagation Applying BP to graphs with loops (cycles) can give the wrong answer, because it overcounts evidence loudy prinkler ain etgrass omputes all N marginals in O(N) parallel updates P- In practice, often works well (e.g., error correcting codes) P- hy Loopy BP? Factor graphs Kschischang0 e can compute exact answers by converting a loopy graph to a junction tree and running BP (see later). However, the resulting Jtree has nodes with O(K w+ ) states, so inference takes O(N K w+ ) time [w=clique size of triangulated graph]. e can apply BP to the original graph in O(N K ) time [ = clique size of original graph]. To apply BP to a graph with non pairwise potentials, it is simpler to use factor graphs. X X Bayes net X X X X X X Markov net X X X X Pairwise Markov net X X X X X X P-5 Bipartite graph P-6 Loopy BP (see MacKay PDF) Dashed messages are products of same color solid messages (and factor) f f f q x f q x f q y f q y f r f x = f r f x r f y r f y r f z x y r f x = f q x f q z f = (empty ) z um-product vs max-product um-product computes marginals using this rule Max-product computes max marginals using the rule f ame algorithm on different semirings: (+,x,0,) and (max,x,-,) hafer90,bistarelli97,goodman99,aji00 P-7 P-8

4 Viterbi decoding ompute most probable explanation (MPE) of observed data Hidden Markov Model (HMM) X X X hidden Viterbi algorithm for HMMs un max forwards algorithm, keeping track of most probable predecessor for each state Y Y Y observed Pointer traceback Tomato an produce N-best list (most probable configurations) in O(N T K ) time Forney7,Nilsson0 P-9 P-0 Loopy Viterbi BP speedup tricks Use max-product to compute/ approximate If there are no ties and the max-marginals are exact, then This method does not use traceback, so can be used with distributed/ loopy BP e can break ties, and produce N most-probable configurations, by asserting that certain assignments are disallowed, and rerunning Yanover0 ometimes we can reduce the time to compute a message from O(K ) to O(K) If (x i,x j ) = exp( f(x i ) f(x j ) ), then um-product in O(K log K) time [exact FFT] or O(K) time [approx] Felzenszwalb0/0,Movellan0,deFreitas0 Max-product in O(K) time [distance transform] For general (discrete) potentials, we can dynamically add/delete states to reduce K oughlan0 ometimes we can speedup convergence by Using a better message-passing schedule (e.g., along ainwright0 embedded spanning trees) Using a multiscale method Felzenszwalb0 P- P- Junction/ join/ clique trees Part II: um Product Algorithm and (Loopy) Belief Propagation Not tested material To perform exact inference in an arbitrary graph, convert it to a junction tree, and then perform belief propagation. A jtree is a tree whose nodes are sets, and which has the Jtree property: all sets which contain any given variable form a connected graph (variable cannot appear in disjoint places) moralize Make jtree Maximal cliques = { {,,}, {,,} } eparators = { {,,} Å {,,} = {,} } P-

5 B G Making a junction tree GM D B D lique potentials A F moralize A F E E Each model clique potential gets assigned to one Jtree clique potential {a,b,c} Jensen9 Max spanning tree {b,c,e} {b,d} {b,e,f} Jtree {b,d} ij = i Å j {b,e,f} {b,c,e} {a,b,c} Jgraph P-5 Find max cliques A B Triangulate (order f,d,e,c,b,a) E GT D F Each observed variable assigns a delta function to one Jtree clique potential If we observe =w *, set E(w)= (w,w * ), else E(w)= quare nodes are factors P-6 eparator potentials BP on a Jtree eparator potentials enforce consistency between neighboring cliques on common variables. A Jtree is a MF with pairwise potentials. Each (clique) node potential contains PDs and local evidence. Each edge potential acts like a projection function. e do a forwards (collect) pass, then a backwards (distribute) pass. The result is the Hugin/ hafer-henoy algorithm. quare nodes are factors P-7 P-8 Initial clique potentials contain PDs and evidence Message from clique to separator marginalizes belief (projects onto intersection) [remove c] P-9 P-0 5

6 eparator potentials gets marginal belief from their parent clique. Message from separator to clique expands marginal [add w] P- P- P- oot clique has seen all the evidence P- Marginalize out w and exclude old evidence (e c, e r ) ombine upstream and downstream evidence P-5 P-6 6

7 Add c and exclude old evidence (e c, e r ) ombine upstream and downstream evidence P-7 P-8 Partial beliefs Hugin algorithm Hugin = BP applied to a Jtree using a serial protocol ollect Distribute i i Evidence on now added here The beliefs / messages at intermediate stages (before finishing both passes) may not be meaningful, because any given clique may not have seen all the model potentials/ evidence (and hence may not be normalizable). This can cause problems when messages may fail (eg. ensor nets). One must reparameterize using the decomposable model to ensure meaningful partial beliefs. Paskin0 P-9 ij j quare nodes are separators P-0 ij j 7