Lauritzen-Spiegelhalter Algorithm

Lauritzen-Spiegelhalter AlgorithmLauritzen-Spiegelhalter Algorithm

Probabilistic Inference Probabilistic Inference In Bayes NetworksIn Bayes Networks

Haipeng GuoHaipeng Guo Nov. 08, 2000Nov. 08, 2000

KDD Lab, CIS Department, KSUKDD Lab, CIS Department, KSU

Presentation OutlinePresentation Outline

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

• 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 NetworksBayes Networks

Bayes NetworksBayes Networks

Probabilistic Inference in Bayes NetworksProbabilistic 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 NetworksProbabilistic 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 NetworksProbabilistic 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

• 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 AlgorithmLauritzen-Spiegelhalter Algorithm

Lauritzen-Spiegelhalter AlgorithmLauritzen-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: CliquesL-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 DE

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

Clique 2: {B, D, E}

Step 1: Building the tree of cliquesStep 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 parents2. 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

Step 1.1: MoralizationStep 1.1: Moralization

Input: G - the DAG of a Bayes Network,

Output: Gm - the Moral graph relative to GAlgorithm: “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: TriangulationStep 1.2: Triangulation

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 CliquesStep 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

G5

A1

D8

B2 E3

C4

H7

F6

Step 1.3: Identify CliquesStep 1.3: Identify Cliques

D8

C4

A1

B2

E3

C4

B2

G5E3

C4

G5

F6

E3

G5

H7

C4

Clique 6Clique 5

Clique 4

Clique 3

Clique 1

Clique 2

order them according to their highest labeled nodes according to order

Step 1.4: Build tree of CliquesStep 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 CliquesStep 1.4: Build tree of Cliques

Clq6(Clq5) = P(H|C,G)

(Clq2) = P(D|C)

D8

C4

A1

B2

E3

C4

B2

G5E3

C4

G5

F6

E3

G5

H7

C4

Clq5

Clq4

Clq3

Clq1

Clq2

Clq1

Clq2

Clq3

Clq4 Clq5

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

Ri: Residual nodes Si: Separator nodes

(Clqi): potential probability of Clique i

Step 1: ConclusionStep 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 Clqi

2. Si

3. Ri

4. (Clqi)

Step 2: Computation inside the join treeStep 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 treeStep 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 treeStep 2: Message passing in the join tree

Clq1

Clq2

Clq3

Clq4 Clq5

Clq6

Upward messages

Downward messages

• ’ (Clqi ’) is modified as the

messages passing is going

• P’ (Clqi’) is computed as the

messages passing is going

Step 2: Message passing in the join treeStep 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

vwClqw

ii

ClqPvP,'

)'(':)('

L-S Algorithm:Computational Complexity L-S Algorithm:Computational Complexity AnalysisAnalysis

1. 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 L-S Algorithm:Computational Complexity AnalysisAnalysis

2. 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

• 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 L-S Algorithm:Computational Complexity AnalysisAnalysis

• 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 methodsL-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 algorithms2. Monte Carlo techniques3. Heuristic algorithms4. Parallel algorithms5. Special case algorithms

L-S Algorithm: DemoL-S Algorithm: Demo

• Laura works on step 1• Ben works on step 2