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

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

Bayes Networks 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

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?

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

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

Lauritzen-Spiegelhalter Algorithm 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 , 1988.

Lauritzen-Spiegelhalter Algorithm 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.

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 A B C D E Clique 1: {A, B, C, D} Clique 2: {B, D, E}

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: Construct a moral graph from the DAG of a Bayes network by marrying parents Add arcs to the moral graph to form a triangulated graph and create an order  of all nodes using Maximum Cardinality Search Identify cliques from the triangulated graph and order them according to order  Connect these cliques to build a join tree

Step 1.1: Moralization Input: G - the DAG of a Bayes Network, Output: Gm - 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 B E G C H F A D B E G C H F

Step 1.2: Triangulation A D B E G C H F Input: Gm - the Moral graph Output: Gu – a perfect ordering  of the nodes and the triangulated graph of Gm Algorithm: 1. Use Maximum Cardinality Search to create a perfect ordering of the nodes 2. Use Fill-in Computation algorithm to triangulate Gu A D B E G C H F A1 D8 B2 E3 G5 C4 H7 F6

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

Step 1.3: Identify Cliques A1 B2 E3 G5 F6 H7 Clique 6 Clique 5 Clique 4 Clique 3 Clique 1 Clique 2 G5 A1 D8 B2 E3 C4 H7 F6 order them according to their highest labeled nodes according to order 

Step 1.4: Build tree of Cliques Input: a list of cliques of the triangulated graph Gu Output: Create a tree of cliques, compute Separators nodes Si,Residual nodes Ri and potential probability (Clqi) for all cliques Algorithm: 1. Si = Clqi (Clq1  Clq2 … Clqi-1) 2. Ri = Clqi - Si 3. If i >1 then identify a j < i such that Clqj is a parent of Clqi 4. Assign each node v to a unique clique Clqi that v  c(v)  Clqi 5. Compute (Clqi) = f(v) Clqi =P(v|c(v)) {1 if no v is assigned to Clqi} 6. Store Clqi , Ri , Si, and (Clqi) at each vertex in the tree of cliques

Step 1.4: Build tree of Cliques Ri: Residual nodes Si: Separator nodes (Clqi): potential probability of Clique i D8 C4 A1 B2 E3 G5 F6 H7 Clq5 Clq4 Clq3 Clq1 Clq2 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 CG EG C (Clq5) = P(H|C,G) Clq6 (Clq2) = P(D|C)

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: A clique Clqi Si Ri (Clqi)

Step 2: Computation inside the join tree In step 2, we start from copying a copy of the Permanent Tree of Cliques to Clqi’, Si’,, Ri’ and ’ (Clqi ’) and P’ (Clqi’) , leave P’ (Clqi’) unassigned at first. Then we compute the prior probability P’ (Clqi’) 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’ (Clqi’) 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

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(’ (Clqi ’) and P’ (Clqi’) ) 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

Step 2: Message passing in the join tree Clq1 Clq2 Clq3 Clq4 Clq5 Clq6 Upward  messages ’ (Clqi ’) is modified as the  messages passing is going Downward  messages P’ (Clqi’) is computed as the  messages passing is going

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

L-S Algorithm:Computational Complexity Analysis Computations in the Algorithm which creates the permanent Tree of Cliques --- O(nrm) 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 (Clqi) – O(nrm) --- n = number of variables; m = the maximum number of variables in a clique; r = maximum number of alternatives for a variable

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

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

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: Approximate algorithms Monte Carlo techniques Heuristic algorithms Parallel algorithms Special case algorithms

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