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Lauritzen-Spiegelhalter Algorithm Probabilistic Inference In Bayes Networks Haipeng Guo Nov. 08, 2000 KDD Lab, CIS Department, KSU

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Presentation Outline Bayes Networks Probabilistic Inference in Bayes Networks L-S algorithm Computational Complexity Analysis Demo

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A Bayes network is a directed acyclic graph with a set of conditional probabilities The nodes represent random variables and arcs between nodes represent conditional dependence of the nodes. Each node contains a CPT(Conditional Probabilistic Table) that contains probabilities of the node being a specific value given the values of its parents Bayes Networks

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Probabilistic Inference in Bayes Networks Probabilistic Inference in Bayes Networks is the process of computing the conditional probability for some variables given the values of other variables (evidence). P(V=v| E=e): Suppose that I observe e on a set of variables E(evidence ), what is the probability that variable V has value v, given e?

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Probabilistic Inference in Bayes Networks Example problem: Suppose that a patient arrives and it is known for certain that he has recently visited Asia and has dyspnea. What’s the impact that this has on the probabilities of the other variables in the network ? Probability Propagation in the network

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Probabilistic Inference in Bayes Networks The problem of exact probabilistic inference in an arbitrary Bayes network is NP-Hard.[Cooper 1988] NP-Hard problems are at least as computational complex as NP-complete problems No algorithms has ever been found which can solve a NP-complete problem in polynomial time Although it has never been proved whether P = NP or not, many believe that it indeed is not possible. Accordingly, it is unlikely that we could develop an general-purpose efficient exact method for propagating probabilities in an arbitrary network

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L-S algorithm is an efficient exact probability inference algorithm in an arbitrary Bayes Network S. L. Lauritzen and D. J. Spiegelhalter. Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society, Lauritzen-Spiegelhalter Algorithm

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L-S algorithm works in two steps: First, it creates a tree of cliques(join tree or junction tree), from the original Bayes network; Then, it computes probabilities for the cliques during a message propagation and the individual node probabilities are calculated from the probabilities of cliques.

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L-S Algorithm: Cliques An undirected graph is compete if every pair of distinct nodes is adjacent a clique W of G is a maximal complete subset of G, that is, there is no other complete subset of G which properly contains W AB CD E Clique 1: {A, B, C, D} Clique 2: {B, D, E}

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Step 1: Building the tree of cliques In step 1, we begin with the DAG of a Bayes Network, and apply a series of graphical transformation that result in a join tree: 1.Construct a moral graph from the DAG of a Bayes network by marrying parents 2.Add arcs to the moral graph to form a triangulated graph and create an order of all nodes using Maximum Cardinality Search 3.Identify cliques from the triangulated graph and order them according to order 4.Connect these cliques to build a join tree

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Step 1.1: Moralization Input: G - the DAG of a Bayes Network, Output: G m - the Moral graph relative to G Algorithm: “marry” the parents and drop the direction (add arc for every pair of parents of all nodes) A D BE G C H F A D BE G C H F

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Step 1.2: Triangulation Input: G m - the Moral graph Output: G u – a perfect ordering of the nodes and the triangulated graph of G m Algorithm: 1. Use Maximum Cardinality Search to create a perfect ordering of the nodes 2. Use Fill-in Computation algorithm to triangulate G u A D BE G C H F A1A1 D8D8 B2B2E3E3 G5G5 C4C4 H7H7 F6F6

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Step 1.3: Identify Cliques Input: G u and a ordering of the nodes Output: a list of cliques of the triangulated graph G u Algorithm: Use Cliques-Finding algorithm to find cliques of a triangulated graph then order them according to their highest labeled nodes according to order

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G5G5 A1A1 D8D8 B2B2E3E3 C4C4 H7H7 F6F6 Step 1.3: Identify Cliques D8D8 C4C4 A1A1 B2B2 E3E3 C4C4 B2B2 G5G5 E3E3 C4C4 G5G5 F6F6 E3E3 G5G5 H7H7 C4C4 Clique 6 Clique 5 Clique 4 Clique 3 Clique 1 Clique 2 order them according to their highest labeled nodes according to order

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Step 1.4: Build tree of Cliques Input: a list of cliques of the triangulated graph G u Output: Create a tree of cliques, compute Separators nodes S i,Residual nodes R i and potential probability (Clq i ) for all cliques Algorithm: 1. S i = Clq i (Clq 1 Clq 2 … Clq i-1 ) 2. R i = Clq i - S i 3. If i >1 then identify a j < i such that Clq j is a parent of Clq i 4. Assign each node v to a unique clique Clq i that v c(v) Clq i 5. Compute (Clq i ) = f(v) Clqi =P(v|c(v)) {1 if no v is assigned to Clq i } 6. Store Clq i, R i, S i, and (Clq i ) at each vertex in the tree of cliques

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Step 1.4: Build tree of Cliques Clq6 (Clq5) = P(H|C,G) (Clq2) = P(D|C) D8D8 C4C4 A1A1 B2B2 E3E3 C4C4 B2B2 G5G5 E3E3 C4C4 G5G5 F6F6 E3E3 G5G5 H7H7 C4C4 Clq5 Clq4 Clq3 Clq1 Clq2 Clq1 Clq2 Clq3 Clq4Clq5 Clq6 Clq3 = {E,C,G} R3 = {G} S3 = { E,C } Clq1 = {A, B} R1 = {A, B} S1 = {} Clq2 = {B,E,C} R2 = {C,E} S2 = { B } Clq4 = {E, G, F} R4 = {F} S4 = { E,G } Clq5 = {C, G,H} R5 = {H} S5 = { C,G } Clq6 = {C, D} R5 = {D} S5 = { C} (Clq1) = P(B|A)P(A) (Clq2) = P(C|B,E) (Clq3) = 1 (Clq4) = P(E|F)P(G|F)P(F) AB BEC ECG EGF CGH CD B EC CGEG C R i : Residual nodesS i : Separator nodes (Clq i ): potential probability of Clique i

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Step 1: Conclusion In step 1, we begin with the DAG of a Bayes Network, and apply a series of graphical transformation that result in a Permanent Tree of Cliques. Stored at each vertex in the tree are the following: 1.A clique Clq i 2.S i 3.R i 4. (Clq i )

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Step 2: Computation inside the join tree In step 2, we start from copying a copy of the Permanent Tree of Cliques to Clq i ’, S i ’,, R i ’ and ’ (Clq i ’ ) and P ’ ( Clq i ’ ), leave P ’ ( Clq i ’ ) unassigned at first. Then we compute the prior probability P ’ ( Clq i ’ ) using the same updating algorithm as the one to determine the conditional probabilities based on instantiated values. After initialization, we compute the posterior probability P ’ ( Clq i ’ ) again with evidence by passing and messages in the join tree When the probabilities of all cliques are determined, we can compute the probability for each variable from any clique containing the variable

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Step 2: Message passing in the join tree The message passing process consists of first sending so-called messages from the bottom of the tree to the top, then sending messages from the top to the bottom, modifying and accumulating node properties( ’ (Clq i ’ ) and P ’ ( Clq i ’ ) ) along the way The message upward is a summed product of all probabilities below the given node The messages downward is information for updating the node prior probabilities

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Step 2: Message passing in the join tree Clq1 Clq2 Clq3 Clq4Clq5 Clq6 Upward messages Downward messages ’ (Clq i ’ ) is modified as the messages passing is going P ’ ( Clq i ’ ) is computed as the messages passing is going

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Step 2: Message passing in the join tree When the probabilities of all cliques are determined, for each vertex Clq i and each variable v Clq i, do

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L-S Algorithm:Computational Complexity Analysis 1.Computations in the Algorithm which creates the permanent Tree of Cliques --- O(nr m ) Moralization – O(n) Maximum Cardinality Search – O(n+e) Fill-in algorithm for triangulating – O(n+e) ---- the general problem for finding optimal triangulation (minimal fill-in) is NP-Hard, but we are using a greedy heuristic Find Cliques and build join tree – O(n+e) --- the general problem for finding minimal Cliques from an arbitrary graph is NP-Hard, but our subject is a triangulated graph Compute (Clq i ) – O(nr m ) --- n = number of variables; m = the maximum number of variables in a clique; r = maximum number of alternatives for a variable

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L-S Algorithm:Computational Complexity Analysis 2.Computations in the updating Algorithm --- O(pr m ) Computation for sending all messages --- 2pr m Computation for sending all messages --- pr m Computation for receiving all messages --- pr m ---- p = number of vertices in the tree of cliques L-S algorithm has a time complexity of O(p r m ), in the worst case it is bounded below by 2 m, i.e. (2 m )

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L-S Algorithm:Computational Complexity Analysis It may seem that we should search for a better general- purpose algorithm to perform probability propagation But in practice, most Bayes networks created by human hands should often contain small clusters of variables, and therefore a small value of m. So L-S algorithm works efficiently for many application because networks available so far are often sparse and irregular. L-S algorithm could have a very bad performance for more general networks

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L-S Algorithm: Alternative methods Since the general problem of probability propagation is NP- Hard, it is unlikely that we could develop an efficient general- purpose algorithm for propagating probabilities in an arbitrary Bayes network. This suggests that research should be directed towards obtaining alternative methods which work in different cases: 1.Approximate algorithms 2.Monte Carlo techniques 3.Heuristic algorithms 4.Parallel algorithms 5.Special case algorithms

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L-S Algorithm: Demo Laura works on step 1 Ben works on step 2

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