COMP538: Introduction to Bayesian Networks
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1 COMP538: Introduction to ayesian Networks Lecture 4: Inference in ayesian Networks: The VE lgorithm Nevin L. Zhang Department of Computer Science and Engineering Hong Kong University of Science and Technology Fall 2008 Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
2 Objective Discuss the variable elimination (VE) algorithm for inference in ayesian networks Reading: Zhang and Guo, Chapter 4 Reference: Zhang and Poole (1994, 1996 (first few sections)); Dechter (1996) Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
3 Outline Posterior Probability Queries 1 Posterior Probability Queries Types of queries 2 The Variable Elimination Inference lgorithm Naive lgorithm Principle via Example Factorization and Variable Elimination The VE lgorithm 3 Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
4 Posterior Probability Queries Queries about posterior probability Posterior queries: Given: The values of some variables. Task: Compute the posterior probability distributions of other variables? MP and MPE queries to be discussed later. Example: oth John and Mary called to report alarm. What is the probability of burglary? Formally, what is the posterior probability distribution P( J=y, M=y)? General form of query: P(Q E=e)? Q is a list of query variables, usually one. E is a list of evidence variables,and e is the corresponding list observed values. Note: old capital letters denote sets of variables. Inference refers to the process of computing the answer to a query. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
5 Posterior Probability Queries Types of queries Diagnostic and Predictive Inference P() E P(E) P(, E) Semantically, four types of queries: Diagnostic inference: From effects to causes. P( M=y) Machine malfunctions. What is wrong? J P(J ) M P(M ) Predictive/Causal inference: From causes to effects. P(M =y) If I hand out candies, will the students like this course better? Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
6 Inter-causal inference Posterior Probability Queries Types of queries Inter-causal inference: etween causes of a common effect. Example: P( =y, E=y) Explaining away: P() E P(E) P(=y =y)<p(=y =y, E=y) P(, E) Earthquake explains away = y. J P(J ) M P(M ) P(=y =y)>p(=y =y, E=n) Exercise: Verify the inequalities. Note: Difficult with logic rules rules: = y = y(0.8). E = y = y(0.9). Fact: E = y Conclusion: = y(0.72). Wrong! Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
7 Mixed Inference Posterior Probability Queries Types of queries P() E P(E) P(, E) Mixed inference: Combining two or more of the above. P( J=y, E =Y ) (Simultaneous use of diagnostic and causal inferences) J P(J ) M P(M ) P( J=y, E=n) (Simultaneous use of diagnostic and inter-causal inferences) ll those types can be handled in the same way. In logic inference, different query types are handled differently: Predictive inference: deduction. Diagnostic inference: abduction. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
8 Outline The Variable Elimination Inference lgorithm 1 Posterior Probability Queries Types of queries 2 The Variable Elimination Inference lgorithm Naive lgorithm Principle via Example Factorization and Variable Elimination The VE lgorithm 3 Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
9 The Variable Elimination Inference lgorithm naive inference algorithm Naive lgorithm Naive algorithm for computing P(Q E = e) in a ayesian network: Example Get joint probability distribution P(X) over the set X of all variables by multiplying conditional probabilities. Marginalize P(Q,E) = X Q EP(X), P(E) = P(Q, E) Q Condition: P(Q E = e) = P(Q,E = e) P(E = e) P(, J, M) = E, P(, E,, J, M), P(J, M) = P(, J, M). P( J=y, M=y) = P(,J=y,M=y) P(J=y,M=y)) Not making use of the factorization,exponential complexity. Key issue: How to exploit the factorization to avoid exponential complexity? Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
10 The Variable Elimination Inference lgorithm Principle Through Example Principle via Example Network: P(), P( ), P(C ), P(D C). Query: P(D)? Computation: P(D) = = C = C = C C D,,C P(,, C, D) P()P( )P(C )P(D C) (1) P(C )P(D C) P()P( ) P(D C) P(C ) P()P( ) (2) Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
11 The Variable Elimination Inference lgorithm Principle Through Example Principle via Example Complexity Number of numerical summations: Use (1): Use (2): Exercise: How about numerical multiplications? Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
12 The Variable Elimination Inference lgorithm Principle via Example Principle Through Example Rewrite expression (2) into an algorithm: Let F = {P(), P( ), P(C ), P(D C)} Remove from F all the functions that involve, create a new function by ψ 1 () = P()P( ). put the new function onto F : F = {ψ 1 (), P(C ), p(d C)}. Remove from F all the functions that involve, create a new function by ψ 2 (C) = P(C )ψ 1 (). put the new function onto F : F = {ψ 2 (C), p(d C)}. Remove from F all the function that involve C, create a new function by ψ 3 (D) = C P(D C)ψ 2 (C). Return ψ 3 (D) (which is exactly P(D)). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
13 Factorization The Variable Elimination Inference lgorithm Factorization and Variable Elimination factorization of a joint distribution is a list of functions whose product is the joint distribution. Functions on the list are called factors. N gives a factorization of a joint probability: n P(X 1, X 2,..., X n ) = P(X i pa(x i )). Example: i=1 C D E F This N factorizes P(,, C, D, E, F) into the following list of factors: P(), P(), P(C), P(D, ), P(E, C), P(F D, E). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
14 The Variable Elimination Inference lgorithm Eliminating a variable Factorization and Variable Elimination Consider a joint distribution P(Z 1, Z 2,..., Z m ) Eliminating Z 1 from P means to compute P(Z 2,..., Z m ) = Z 1 P(Z 1, Z 2,...,Z m ). The complexity is exponential in m. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
15 The Variable Elimination Inference lgorithm Factorization and Variable Elimination Eliminating a variable Now suppose we have factorization: P(Z 1, Z 2,...,Z m ) = f 1 f 2... f n Obtaining a factorization of P(Z 2,...,Z m ) could be done with much less computation: Procedure eliminate(f, Z): Inputs: F list of functions; Z variable. Output: nother list of functions. 1 Remove from the F all the functions, say f 1,..., f k, that involve Z, 2 Compute new function g = k i=1 f i. 3 Compute new function h = Z g. 4 dd the new function h to F. 5 Return F. Z k i=1 f i can be much cheaper than Z P(Z 1, Z 2,...,Z m ). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
16 The Variable Elimination Inference lgorithm Eliminating a variable Factorization and Variable Elimination Theorem (4.1) Suppose F is a factorization of a joint probability distribution P(Z 1, Z 2,...,Z m ).Then eliminate(f, Z 1 ) is a factorization of the marginal probability distribution P(Z 2,...,Z m ). Proof: Suppose F consists of factors f 1, f 2,..., f n. Suppose Z 1 appears in and only in factors f 1, f 2,..., f k. P(Z 2,..., Z m ) = Z 1 P(Z 1, Z 2,..., Z m ) = n f i = k n f i f i Z 1 i=1 Z 1 i=1 i=k+1 n = [ f i ][ k n f i ] = [ f i ]h. Q.E.D i=k+1 Z 1 i=1 i=k+1 Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
17 The Variable Elimination Inference lgorithm Observed variable instantiation The VE lgorithm Function h(x, Y ). Suppose X is observed and X=0. X \ Y Instantiating X in h (to its observed value) resulting a function g(y) = h(x = 0, Y) of Y only: Y Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
18 The Variable Elimination Inference lgorithm The VE lgorithm The Variable Elimination lgorithm Procedure VE(F,Q,E,e, ρ) //for computing P(Q E=e): Inputs: F The list of CPTs in a N; Q list of query variables; E list of observed variables; e Observed values; ρ Ordering of variables / Q E(Elimination ordering). Output: P(Q E=e). 1 While ρ is not empty, 1 Remove the first variable Z from ρ, 2 Call eliminate(f, Z). Endwhile 2 Set h = product of all the factors in F. 3 Instantiate observed variables in h to their observed values. 4 Return h(q)/ Q h(q). // Re-normalization Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
19 Example The Variable Elimination Inference lgorithm The VE lgorithm Query: P( F = 0)? C D E F Elimination ordering: ρ C, E,, D Initial factorization: F = {P(), P(), P(C), P(D, ), P(E, C), P(F D, E)} Inference process: Step 1, eliminate C: F = {P(), P(), P(D, ), P(F D, E), ψ 1 (, E)} where ψ 1 (, E) = C P(C)P(E, C). Step 1, eliminate E: F = {P(), P(), P(D, ), ψ 2 (, D, F)} where ψ 2 (, D, F) = E P(F D, E)ψ 1(, E). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
20 The Variable Elimination Inference lgorithm Example (cont d) The VE lgorithm Continued from previous slide Step 1, eliminate : F = {P(), ψ 3 (, D, F)} where ψ 3 (, D, F) = P()P(D, )ψ 2(, D, F) Step 1, eliminate D: where ψ 4 () = D ψ 3(, D, F) Step 2: h(, F) = P()ψ 4 (, F). Step 3: h() = h(, F = 0). Step 4: P( F=0) = h() F = {P(), ψ 4 (, F)} h(). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
21 The Variable Elimination Inference lgorithm The VE lgorithm TheNevin lastl. step Zhang is (HKUST) called renormalization. ayesian Networks Fall / 43 The Variable Elimination lgorithm Theorem (4.2) The output of VE(F,Q,E,e, ρ) is P(Q E=e). Proof: y repeatedly applying Theorem 4.1, we conclude that,after the while-loop, F is a factorization of P(Q,E). Hence, after step 2, h is: h(q,e) = P(Q,E). fter step 3, h is: h(q) = P(Q, E=e). Consequently, h(q) Q h(q) = P(Q,E=e) P(Q,E=e) = = P(Q E=e). Q.E.D P(Q,E=e)) P(E=e)) Q
22 Modification The Variable Elimination Inference lgorithm The VE lgorithm Procedure VE(F, Q, E, e, ρ) 1 Instantiate observed variables in all functions. 2 While ρ is not empty, 1 Remove the first variable Z from ρ, 2 Call eliminate(f, Z). Endwhile 3 Set h = the multiplication of all the factors on F. 4 Return h(q)/ Q h(q). Exercises: Formally show the correctness of this version of VE. Explain why it is more efficient that the version given earlier. Note: This algorithm was first described in Zhang and Poole (1994). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
23 Outline Complexity of the VE lgorithm 1 Posterior Probability Queries Types of queries 2 The Variable Elimination Inference lgorithm Naive lgorithm Principle via Example Factorization and Variable Elimination The VE lgorithm 3 Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
24 Complexity of the VE lgorithm Measuring the Complexity of One Step For any variable, let w(x) be the number of possible values of X. Complexity of eliminate: t step 2, a new function g is constructed. The size of g= {w(x) : X appears in one of the functions that involve Z.}. The size is a good and nature measurement of the complexity of eliminating Z. (ccurate operation counts are difficult.) We call the size of g the cost of eliminating z from F and denote it by c(z). In the previous example, assume all variables are binary. The cost of eliminating C is: 8 The cost of eliminating E is: 16 The cost of eliminating is: 16 The cost of eliminating D is: 8 Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
25 Complexity of the VE lgorithm Measuring the Complexity of the VE lgorithm Complexity of VE: Suppose the elimination ordering is: Z 1, Z 2,..., Z m. The cost of VE is defined to be: m c(z i ) i=1 Complexity in the previous example: Cost of VE is: = 36. Often, one term dominates all others. The term usually referred to as maximum clique size. We will see the reason behind this terminology later. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
26 Complexity of the VE lgorithm Determining Complexity of Inference Determining Complexity of Inference from Network Structure It is often desirable to know the complexity of inference beforehand. In the next few slides, we show how the complexity of VE can easily be determined from network structure. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
27 Complexity of the VE lgorithm Structural Graph of Factorization Determining Complexity of Inference from Network Structure Given a list F of function, the structural graph of F is an undirected graph obtained as follows: Example: For any two variables X and Y, connect them iff they appear in the same factor. F = {P(), P(T ), P(S), P(L S), P( S), P(R T, L), P(X R), P(D R, )} The structural graph of F is: S T L X R D Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
28 Moral Graph of DG Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure The moral graph m(g) of a DG is the undirected graph obtained from G by Marrying the parents of each node (i.e adding an edge between each pair of parents), and Dropping all directions. S S T L T L X R D X R D Note: If F is the list of CPTs of a N, then the structural graph of F is simply the moral graph of the N. F = {P(), P(T ), P(S), P(L S), P( S), P(R T, L), P(X R), P(D R, )} Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
29 Complexity of the VE lgorithm Cost of Eliminating One Variable Determining Complexity of Inference from Network Structure For any vertex Z in an undirected graph, let adj(z) be the set of all neighbors of Z. Fact 1: If G is the structural graph of F, the cost of eliminating Z from F is given by Why?Recall c(z) = w(z) X adj(z) w(x). 1 Remove from the F all the functions, say f 1,..., f k, that involve Z, 2 Compute new function g = k i=1 f i Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
30 Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Cost of Eliminating One Variable F = {P(), P(T ), P(S), P(L S), P( S), P(R T, L), P(X R), P(D R, )} The structural graph of F is: S T L X R D Eliminating T: Needs to compute: P(T )P(R T, L) Cost: c(t) = w(t)w()w(r)w(l) adj(t) = {, R, L}. So, c(t) = w(t) X adj(t) w(x). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
31 Complexity of the VE lgorithm Eliminating Vertex from Graph Determining Complexity of Inference from Network Structure Eliminating a vertex Z from an undirected graph G means: dding edges so that all nodes in adj(z) are pairwise adjacent,and Removing Z and its incident edges. Denote the result graph by eliminate(g, Z). Example: eliminate(g, T) is S S T L L X R D X R D Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
32 Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Elimination in Factorization and Elimination in Graph Fact 2: If G is the structural graph of F, then eliminate(g, Z) is the structural graph of eliminate(f, Z). Example: eliminate(f, T) = {P(), P(S), P(L S), P( S), P(X R), P(D R, ), ψ(, L, R)}, where ψ(, L, R) = T P(T )P(R T, L). eliminate(g, T) is S S T L L R R D D X X We see that eliminate(g, T) is the graph for eliminate(f, T). Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
33 Complexity of the VE lgorithm Determining the Complexity of VE Determining Complexity of Inference from Network Structure Fact 1 and Fact 2 allow us to determine the complexity of VE by manipulating graphs. Z 1 Z 2 Z 3 F 1 F 2 F 3... G 1 G 2 G 3... c(z 1 ) c(z 2 ) c(z 3 ) There are no numerical calculations in the process. It is fast. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
34 Complexity of the VE lgorithm Determining the Complexity of VE Procedure costve(n, E, ρ) Inputs: N ayesian network structure. E Set of observed variables. ρ n elimination ordering. Output: complexity of VE. 1 Compute moral graph G of N. 2 Remove from G all nodes in E. // Structural graph of F after step 1 of VE 3 C = 0. 4 While ρ is not empty, 1 Remove the first variable Z from ρ, 2 C += w(z) X adj(z) w(x). 3 eliminate(g, Z). 5 Return C Determining Complexity of Inference from Network Structure Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
35 Example of costve Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Example 1:, X, D,, S, L, T, R S S S S T L T L T L T L X R D X R D R D R Eliminate: Cost: 22 Eliminate: X Cost: 2 2 Eliminate: D Cost: 23 Eliminate: S Cost: 23 T R L T R L T R R Eliminate: Cost: 23 Eliminate: L Cost: 23 Eliminate: T Cost: 2 2 Eliminate: R Cost: 2 Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
36 Example of costve Complexity of the VE lgorithm Determining Complexity of Inference from Network Structure Example 2: R, L, T, D,, S,, X S S S S T L T L T R D D D D X X X X Eliminate: T Eliminate: D Eliminate: Eliminate: L Cost: 2 6 R Cost: 2 Cost: 2 Cost: 2 S S X Eliminate: Cost: 2 4 X Eliminate: S Cost: 2 3 X Eliminate: Cost: 2 2 X Eliminate: RX Cost: 2 Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
37 Complexity of the VE lgorithm Optimal Elimination Ordering Determining Complexity of Inference from Network Structure Different elimination orderings lead to different costs. The optimal elimination ordering: the one with minimum cost. It is NP-hard to find an optimal elimination ordering (rnborg et al, 1987). The best we can hope for are some heuristics. Will give some heuristics in Lecture 4.1. Nevin L. Zhang (HKUST) ayesian Networks Fall / 43
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