Exact Inference 275 b SP 2 What are
Exact Inference 275 b SP 2
What are they good for? n Diagnosis: P(cause|symptom)=? n Prediction: P(symptom|cause)=? n Classification: n Decision-making (given a cost function) Medicine P(class|data) symptom Bioinformatics Speech recognition Stock market cause Text Classification SP 2 Computer troubleshooting 2
Probabilistic Inference Tasks § Belief updating: § Finding most probable explanation (MPE) § Finding maximum a-posteriory hypothesis § Finding maximum-expected-utility (MEU) decision SP 2 3
Belief Updating Smoking Bronchitis lung Cancer X-ray Dyspnoea P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ? SP 2 4
Belief updating is NP-hard n n n Each sat formula can be mapped to a bayesian network query. Example: (u, ~v, w) and (~u, ~w, y) sat? SP 2 5
Motivation n Given a chain show it works How can we compute P(D)? P(D|A=0)? P(A|D=0)? A n B C D Brute force O(k^4) SP 2 6
Belief updating: A BB D D P(X|evidence)=? P(a, e=0)= P(a|e=0) C C P(a)P(b|a)P(c|a)P(d|b, a)P(e|b, c)= EE “Moral” graph P(a) P(c|a) P(b|a)P(d|b, a)P(e|b, c) Variable Elimination SP 2 7
Bucket elimination Algorithm elim-bel (Dechter 1996) Elimination operator bucket B: bucket C: P(b|a) P(d|b, a) P(e|b, c) P(c|a) B C bucket D: D bucket E: e=0 bucket A: P(a) W*=4 ”induced width” (max clique size) P(a|e=0) SP 2 E A 9
Complexity of elimination The effect of the ordering: A B D C E B E C D D C E B A A “Moral” graph SP 2 10
Handling evidence n Complexity based on adjusted inducedwidth SP 2 11
Relationship with Pearl’s belief propagation in poly-trees “Causal support” “Diagnostic support” Pearl’s belief propagation for single-root query elim-bel using topological ordering and super-buckets for families Elim-bel, elim-mpe, and elim-map are linear for poly-trees. SP 2 12
Relationship with join-tree clustering BCE ADB A cluster is a set of buckets (a “super-bucket”) ABC SP 2 13
Finding Algorithm elim-mpe (Dechter 1996) Elimination operator bucket B: P(b|a) P(d|b, a) P(e|b, c) B bucket C: P(c|a) C bucket D: D bucket E: e=0 bucket A: P(a) W*=4 ”induced width” (max clique size) MPE SP 2 E A 14
Generating the MPE-tuple B: P(b|a) P(d|b, a) P(e|b, c) C: P(c|a) D: E: e=0 A: P(a) SP 2 15
Other tasks and algorithms n MAP and MEU tasks: n n Similar bucket-elimination algorithms - elim-map, elim-meu (Dechter 1996) Elimination operation: either summation or maximization Restriction on variable ordering: summation must precede maximization (i. e. hypothesis or decision variables are eliminated last) Other inference algorithms: n Join-tree clustering n Pearl’s poly-tree propagation n Conditioning, etc. SP 2 16
Conditioning generates the probability tree Complexity of conditioning: exponential time, linear space SP 2 17
Conditioning+Elimination Idea: conditioning until of a (sub)problem gets small SP 2 18
Super-bucket elimination (Dechter and El Fattah, 1996) n n Eliminating several variables ‘at once’ Conditioning is done only in super-buckets SP 2 19
The idea of super-buckets Larger super-buckets (cliques) =>more time but less space Complexity: 1. Time: exponential in clique (super-bucket) size 2. Space: exponential in separator size SP 2 20
OR search space A F B E A 1 0 E 1 0 C F D 0 B D Ordering: A B E C D F C 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 SP 2 21
AND/OR search space A A A F B C B C C E E D E Primal graph D D F DFS tree OR A AND 0 1 OR B B AND 0 OR AND E 0 OR AND C 1 E 0 1 D 0 F 1 0 0 D 1 0 F 1 0 0 0 F 1 0 1 D 1 E C 1 0 D 1 1 0 1 SP 2 0 E 0 1 D F 1 C F 1 0 0 D 1 0 1 F 1 0 C 0 1 D 1 0 F 1 0 D 1 0 F 1 22 0 1
A A F B OR AND 0 1 B B 0 OR C D D F 0 C 1 OR 1 D AND 0 E 0 F 1 0 0 D 1 0 F 1 0 1 A 0 0 F 1 0 D 1 0 C 0 1 D 1 E C 1 AND/OR 0 1 E AND 1 1 0 0 1 D F 0 1 E F 1 0 0 D 1 0 1 0 C 1 0 0 1 1 0 1 0 1 1 D 1 F 0 1 0 D 1 0 0 0 F 1 0 1 1 0 E C 1 F 0 B F E A AND D E OR vs AND/OR OR B C 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 OR 1 0 1 0 1 0 1 0 1 0 1 0 1 SP 2 24
A A F B AND/OR vs. OR OR B C E E C D D F A AND OR AND 0 1 B B 0 OR AND 0 C 1 OR 0 E 0 1 D AND F 1 0 1 E 0 0 D 1 0 F 1 0 0 1 0 F 1 0 D 1 0 C 0 1 D 1 E C 1 AND/OR 1 1 0 0 1 D F 0 1 E F 1 0 0 D 1 0 1 F 1 0 C 0 1 D 1 F 0 1 0 D 1 0 F 1 0 1 AND/OR size: exp(4), OR size exp(6) A 0 B 0 E F 1 0 C D 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 SP 2 1 1 0 0 0 1 OR 1 0 1 0 1 0 1 1 0 25 1 0 1 0 1 0 1 0 1
OR AND E 0 1 B B 0 OR E C D D F 0 C 1 OR 1 D AND 0 E 0 F 1 0 0 D 1 0 F 1 0 1 A 0 0 F 1 0 D 1 0 C 0 1 D 1 E C 1 AND/OR 0 1 E AND 1 1 0 0 1 D F 0 1 E F 1 0 0 D 1 0 1 0 C 1 0 0 1 1 0 1 0 1 1 D 1 F 0 1 0 D 1 0 0 0 F 1 0 1 1 0 E C 1 F 0 B F B C A AND D F B AND/OR vs. OR OR A A No-goods (A=1, B=1) (B=0, C=0) 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 SP 2 1 1 0 0 0 1 OR 1 0 1 0 1 0 1 1 0 26 1 0 1 0 1 0 1 0 1
F B OR AND 0 1 B B 0 OR E 0 C E 1 1 0 D AND 0 1 0 C D F 0 0 1 F 1 AND/OR 0 D 1 0 C 1 1 D F 1 0 0 1 F 1 0 B 1 0 0 1 1 C F 0 D 1 E C 1 F A D E 0 1 OR E D B A AND C E AND/OR vs. OR OR A A (A=1, B=1) (B=0, C=0) 0 1 1 0 1 0 1 0 1 1 0 0 0 1 OR 0 0 1 1 1 0 1 0 1 0 SP 2 1 1 0 1 0 1 0 1 27
OR space vs. AND/OR space width AND/OR space height time(sec. ) nodes backtracks time(sec. ) AND nodes OR nodes 5 10 3. 154 2, 097, 150 1, 048, 575 0. 03 10, 494 5, 247 4 9 3. 135 2, 097, 150 1, 048, 575 0. 01 5, 102 2, 551 5 10 3. 124 2, 097, 150 1, 048, 575 0. 03 8, 926 4, 463 4 10 3. 125 2, 097, 150 1, 048, 575 0. 02 7, 806 3, 903 5 13 3. 104 2, 097, 150 1, 048, 575 0. 1 36, 510 18, 255 5 10 3. 125 2, 097, 150 1, 048, 575 0. 02 8, 254 4, 127 6 9 3. 124 2, 097, 150 1, 048, 575 0. 02 6, 318 3, 159 5 10 3. 125 2, 097, 150 1, 048, 575 0. 02 7, 134 3, 567 5 13 3. 114 2, 097, 150 1, 048, 575 0. 121 37, 374 18, 687 5 10 3. 114 2, 097, 150 1, 048, 575 0. 02 7, 326 3, 663 SP 2 28
AND/OR search tree for graphical models n The AND/OR search tree of R relative to a spanning-tree, T, has: n n Alternating levels of: OR nodes (variables) and AND nodes (values) A Successor function: n n F The successors of OR nodes X are all its consistent values along its path The successors of AND <X, v> are all X child variables in T B C E n n D A solution is a consistent subtree Task: compute the value of the root node A B E OR AND 0 1 OR B B AND 0 OR AND 0 C 1 1 D 0 E 0 F 1 0 0 D 1 0 F 1 0 D 1 0 0 1 SP 2 0 E 0 1 D F 1 C 1 F 1 E C 1 D 0 1 E OR AND C A 0 0 D 1 0 1 F 1 0 C 0 1 D 1 0 F 1 0 1 29
From DFS trees to pseudo-trees (Freuder 85, Bayardo 95) 4 1 6 1 2 3 2 7 5 3 (a) Graph 4 1 2 7 1 7 3 5 4 5 3 4 2 7 6 (b) DFS tree depth=3 (c) pseudo- tree depth=2 SP 2 5 6 6 (d) Chain depth=6 30
From DFS trees to Pseudo-trees OR 1 AND a OR b 2 7 AND a b c OR 3 3 3 5 5 5 DFS tree AND a b c a b c OR 4 4 4 4 4 6 6 6 6 6 AND 1 AND a OR b 3 AND depth = 3 a b c a b c a b c a b c a b c OR OR c a c 5 b c a b c 4 2 4 2 7 6 7 6 a b c a b c a b c SP 2 pseudo- tree depth = 2 31
Finding min-depth backbone trees n n Finding min depth DFS, or pseudo tree is NPcomplete, but: Given a tree-decomposition whose tree-width is w*, there exists a pseudo tree T of G whose depth, satisfies (Bayardo and Mirankar, 1996, bodlaender and Gilbert, 91): m <= w* log n, SP 2 32
Generating pseudo-trees from Bucket trees A (A) A B (AB) B C (AC) (BC) C E (AE) (BE) E F C bucket-C E B A D d: A B C E D F D (BD) (DE) D F (AF) (EF) F Bucket-tree based on d A OR AND E D F Bucket-tree used as pseudo-tree bucket-B ABE bucket-E BDE AEF bucket-F Bucket-tree 1 B B 0 E 1 C 0 1 D 0 F 1 0 1 C OR AND bucket-D 0 0 OR AND AB A AND C bucket-A ABC Induced graph OR B A 0 0 D 1 0 F 1 0 0 0 F 1 0 1 D 1 C E 1 0 D 1 1 0 1 0 C 0 1 D F 1 E F 1 0 0 D 1 0 1 F 1 0 E 0 1 D 1 0 F 1 0 AND/OR search tree SP 2 33 1
AND/OR Search-tree properties (k = domain size, m = pseudo-tree depth. n = number of variables) n n Theorem: Any AND/OR search tree based on a pseudotree is sound and complete (expresses all and only solutions) Theorem: Size of AND/OR search tree is O(n km) Size of OR search tree is O(kn) n Theorem: Size of AND/OR search tree can be bounded by O(exp(w* log n)) n Related to: n When the pseudo-tree is a chain we get an OR space (Freuder 85; Dechter 90, Bayardo et. al. 96, Darwiche 1999, Bacchus 2003) SP 2 35
Tasks and value of nodes n V( n) is the value of the tree T(n) for the task: n n n Counting: v(n) is number of solutions in T(n) Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise. Optimization: v(n) is the optimal solution in T(n) Belief updating: v(n), probability of evidence in T(n). Partition function: v(n) is the total probability in T(n). n Goal: compute the value of the root node recursively using n Theorem: Complexity of AO dfs search is dfs search of the AND/OR tree. n n n Space: O(n) Time: O(n km) Time: O(exp(w* log n)) SP 2 36
DFS algorithm (#CSP example) A A B F B C E E solution D OR C D F 11 A AND 0 5 1 6 OR B 5 B 6 AND 0 OR AND 0 1 0 D 2 2 F 1 D 1 0 4 1 2 F 0 D 1 F 1 D 1 F 0 0 0 1 0 1 1 1 1 0 0 1 0 0 D 0 1 1 F 1 D 2 F 1 0 1 0 1 1 1 0 0 1 1 D 0 1 2 C 1 0 2 2 1 E 1 C 4 E 1 Marginalization C 1 E OR node: operator (summation) 1 0 0 1 operator 0 2 1 2 0 1 node: 0 1 1 0 0 AND Combination (product) 1 0 2 C 1 OR AND 1 2 E 1 4 1 1 1 0 1 F 0 1 1 0 1 D 0 0 1 1 1 0 Value of node = number of solutions below it SP 2 1 F 37
Belief-updating on example A A B B C E E D D C A 0 1 B B 0 1 E 0 D 1 0 E 0 1 C C 1 0 0 0 1 D 1 0 E 0 1 C C 1 1 0 0 D 1 1 0 SP 2 E 0 1 C C 1 0 0 1 D 1 0 0 1 C C 1 0 1 38
A A B B C E E D D C A PA(0) 0 1 B B PB|A(1|0) PB|A(0|0) 0 E 0 1 PB|A(0|1) 1 D PE|AB(1|0, 0) PE|AB(0|0, 0) PA(1) 0 E D 1 C C 0 1 1 E PE|AB(0|0, 1) PE|AB(1|0, 1) 0 PB|A(1|1) D PE|AB(0|1, 0) PE|AB(1|1, 0) 0 1 C C 0 1 E D PE|AB(0|1, 1) PE|AB(1|1, 1) 0 1 C C PD|BC(0|0, 0)× PC|A(0|0) 0 1 0 1 SP 2 0 1 0 1 39
A A B B C E E D D C A P(A=0) 0 B P(B=0|A=0) P(B=1|A=0) 0 E 1 D P(E=0|A=0, B=0) E P(E=1|A=0, B=0) 0 1 P(D=0|B=0, C=0)× P(C=0|A=0) 0 P(E=0|A=0, B=1) 0 1 C C P(D=0|B=0, C=1)× P(C=1|A=0) 1 D P(D=1|B=0, C=0)× P(C=0|A=0) 0 P(D=1|B=0, C=1)× P(C=1|A=0) 0 P(E=1|A=0, B=1) 1 P(D=0|B=1, C=0)× P(C=0|A=0) 1 1 C C P(D=0|B=1, C=1)× P(C=1|A=0) 0 SP 2 0 1 P(D=1|B=1, C=0)× P(C=0|A=0) P(D=1|B=1, C=1)× P(C=1|A=0) 0 1 40
A P(A=0) 0 B P(B=0|A=0) P(B=1|A=0) 0 1 E P(E=0|A=0, B=0) C E P(E=1|A=0, B=0) P(C=0|A=0) 0 1 P(D=0|B=0, C=0) 0 P(E=0|A=0, B=1) P(C=0|A=0) 1 D D 1 P(E=1|A=0, B=1) P(C=1|A=0) 0 P(D=0|B=0, C=1) C P(D=1|B=0, C=0) 0 P(D=1|B=0, C=1) 0 1 P(D=0|B=1, C=0) 1 0 1 D D P(D=0|B=1, C=1) 0 P(C=1|A=0) 1 P(D=1|B=1, C=0) P(D=1|B=1, C=1) 0 1 (d) SP 2 41
From Search Trees to Search Graphs n Any two nodes that root identical subtrees/subgraphs can be merged n Minimal AND/OR search graph: closure under merge of the AND/OR search tree n n Inconsistent subtrees can be pruned too. Some portions can be collapsed or reduced. SP 2 43
A J A F B E AND/OR Tree K C H A AND 0 1 OR B B AND 0 OR AND E D G OR B C E 0 C 1 OR 0 1 E 0 1 D F 0 D 1 F 0 0 1 D F D C 1 G J H K E 0 1 D F F 1 E C D F C 0 D 1 F 0 1 D F AND 0 1 0 1 0 1 0 1 OR G G J J G G J J AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR H H H H K K K K H H H H K K K K AND 0101010101010101010101010101010101010101010101010101010101010101 SP 2 44
A J A F B An AND/OR graph E K A 0 1 OR B B AND 0 OR OR C H AND E D G OR B C E 0 C 1 D 0 1 E 0 1 F D 0 F 1 D 0 0 1 F D C 1 D F AND 0 1 OR G G J J AND 0 1 0 1 OR H H K K AND 01010101 SP 2 F G J H K 1 E C D E 0 1 F D 0 F C 1 0 D 1 F D F 45
Context based caching n n Caching is possible when context is the same context graph A = = J F B C E D G H K parent-separator set in induced pseudo- current Avariable + parents Bconnected to subtree below context(B) = {A, B} E C D F G J H K context(c) = {A, B, C} context(D) = {D} context(F) = {F} SP 2 46
Induced-width of pseudo-trees The induced-width of a pseudo-tree is its induced-width along a dfs order that includes pseudo arcs. • For pseudo-chains induced-width is path-width (yielding path-decomposition) • Contexts are bounded by the induced-width of the pseudo tree. • Min induced-pseudo-width=tree-width A B C A A B B A C C E E E D F F C E B D D D F F Good pseudo-tree Bad pseudo-tree DFS order of both pseudo trees: d=ABCEDF A graph SP 2 48
Induced-width of pseudo-trees The induced-width of a pseudo-tree is its induced-width along a dfs order that includes pseudo arcs. • For pseudo-chains induced-width is path-width (yielding path-decomposition) • Contexts are bounded by the induced-width of the pseudo tr. • Min induced-pseudo-width=tree-width A B C A A B B A C C E E E D F F C E B D D D F F Good pseudo-tree Bad pseudo-tree DFS order of both pseudo trees: d=ABCEDF A graph SP 2 49
A J A F Caching B E K C H A AND context(D)={D} 0 1 OR context(F)={F} B B AND 0 OR AND E D G OR B C E 0 C 1 OR 0 1 E 0 1 D F 0 D 1 F 0 0 1 D F D C 1 G J H K E 0 1 D F F 1 E C D F C 0 D 1 F 0 1 D F AND 0 1 0 1 0 1 0 1 OR G G J J G G J J AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR H H H H K K K K H H H H K K K K AND 0101010101010101010101010101010101010101010101010101010101010101 SP 2 50
A J A F B Caching E context(D)={D} K A 0 1 OR B B 0 OR AND OR 0 C 1 D 0 1 E E 0 1 F D 0 F C H AND E D G context(F)={F} OR B C 1 D 0 0 1 F D C 1 D F AND 0 1 OR G G J J AND 0 1 0 1 OR H H K K AND 01010101 SP 2 F G J H K 1 E C D E 0 1 F D 0 F C 1 0 D 1 F D F 51
Size of minimal AND/OR context graphs Theorem: n n Minimal AND/OR context graph is bounded exponentially by its pseudo-tree induced-width. The tree-width of a pseudo-chain is path-width (pw) n Minimal OR search graph is O(exp(pw*)). Minimal AND/OR graph is O(exp(w*)) n Always, w* ≤ pw*, but pw*<= w* log n n SP 2 52
OR tree vs. OR graph A 0 B 0 E 0 C D 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 OR tree 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 A 0 1 B 0 E 0 C D F 1 0 0 0 1 1 1 0 0 1 1 0 1 OR graph 0 1 1 1 0 1 0 1 0 1 0 1 0 100 1 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 SP 2 54
AND/OR tree vs. AND/OR graph AND/OR tree OR A AND OR AND 1 B B 0 OR AND 0 E 0 C 1 OR 0 E 0 1 D AND F 1 0 0 D 1 0 F 1 0 0 0 F 1 0 1 D 1 E C 1 0 D 1 1 0 OR 0 1 E 0 1 D F 1 C F 1 0 0 D 1 0 0 1 1 OR OR B B 1 D 1 0 F 1 0 1 A AND 0 0 OR OR OR AND 0 A OR AND AND 1 F 1 C 0 0 C 1 1 0 D D E 0 0 1 1 E 1 1 0 F F 1 0 D D 1 0 0 F F 1 0 D D 0 0 1 0 AND 0 0 1 1 F F 1 1 E C 1 AND/OR graph 0 D D 1 0 F F 1 SP 2 C 0 0 1 0 D D E 1 1 1 0 F F 1 0 D D 0 1 0 0 F F 1 1 C 1 0 D D 0 0 1 1 1 0 F F 1 0 D D 1 55 0 F F 1
A A F B B C E E C D D F All four search spaces 0 A 0 B 1 0 E 1 0 C D 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 A 1 0 1 1 0 B 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 0 1 OR B B 1 0 1 0 1 C 0 1 0 1 D 0 1 0 1 F 0 AND E 0 1 OR AND C 0 1 D 0 E AND 0 1 OR B B 1 F 1 0 0 D 1 0 1 F 1 0 0 0 F 1 0 1 D 1 E C 0 D 1 0 0 1 0 E 0 1 D F 1 C 1 F 1 1 A 0 AND OR 1 0 OR 0 1 0 Context minimal OR search graph AND 0 1 E Full OR search tree AND 1 0 0 D 1 Full AND/OR search tree 0 1 F 1 0 C 0 1 D 1 0 F 1 0 AND D 1 E OR 0 F 1 0 OR 1 SP 2 AND 0 C 1 D 0 1 E 0 1 F D E C 0 F D 0 C D 1 E 0 1 F D C 1 F D 0 0 F D 1 F 1 Context minimal AND/OR search graph 56 D F
A A F B B C E E C D D F All four search spaces 0 A 0 B 1 0 E 1 0 C D 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 A 1 0 1 1 0 B 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 1 0 1 E 0 1 0 1 C 0 1 0 1 D 0 1 0 1 F 0 Full OR search tree 1 Context minimal OR search graph Timespace AND 0 1 OR B B AND 0 E 0 C 1 OR AND 0 1 A AND 0 1 OR B B 1 0 AND OR 0 1 D 0 E F 1 0 0 D 1 0 1 F 1 0 0 0 F 1 0 1 D 1 E C 0 D 1 0 1 0 E 0 1 D F 1 C F 1 0 0 D 1 Full AND/OR search tree 0 1 F 1 0 C 0 1 D 1 0 F 1 0 AND D 1 E OR 0 F 1 0 OR 1 SP 2 AND 0 C 1 D 0 1 E 0 1 F D E C 0 F D 0 C D 1 E 0 1 F D C 1 F D 0 0 F D 1 F 1 Context minimal AND/OR search graph 57 D F
AND/OR w-cutset E D C E M A L B G G B L H F G A K M L J A D K H F E M K J B D C L B H E M C K F D K H F J J A A B B K K C G L D F M E H E H J J J 3 -cutset 2 -cutset SP 2 J 1 -cutset 60
AND/OR w-cutset E D C E M A L B K H F G J grahpical model D C A L B K H F G E M J pseudo tree SP 2 D C M A L B K H F G J 1 -cutset tree 61
Searching AND/OR Graphs n AO(i): searches depth-first, cache i-context n i = the max size of a cache table (i. e. number of variables in a context) i=0 i=w* i Space: O(n) Space: O(exp w*) Time: O(exp(w* log n)) Space: O(exp(i) ) Time: O(exp(m_i+i ) SP 2 O(exp w*) 62
w-cutset Trees Over AND/OR space n Definition: n n Theorem: n n T_w is a w-cutset tree relative to backbone tree T, iff T_w is roots T and when removed, yields tree-width w. AO(i) time complexity for backbone T is time O(exp(i+m_i)) and space O(i), m_i is the depth of the T_i tree. Better than w-cutset: O(exp(i+c_i)) when c_i is the number of nodes in T_i SP 2 63
Recursive Conditioning (Darwiche, 1999) n Algorithms that explore the AND/OR search tree and graph. SP 2 64
Decomposition Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Turn Over SP 2 Engine Start 65
Decomposition Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Turn Over SP 2 Engine Start 66
Decomposition Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Start Engine Turn Over SP 2 67
Decomposition Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Start Engine Turn Over LP * SP 2 RP 68
Decomposition Alternator Battery Age Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights LP Engine Start Engine Turn Over * SP 2 RP 69
Causal Network Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Turn Over SP 2 Engine Start 70
Causal Network Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights Engine Turn Over SP 2 Engine Start 71
Case Analysis Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Battery Fuel Line Starter Distributor Battery Power Gas Distributor Battery Power Spark Plugs Gas Gauge Radio Lights Engine Turn Over Gas Gauge Engine Start Radio Case I Lights Engine Turn Over Engine Start Case II SP 2 72
Case Analysis Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Battery Fuel Line Starter Distributor Battery Power Gas Distributor Battery Power Spark Plugs Gas Gauge Radio Lights Engine Turn Over Gas Gauge Engine Start Radio Case I Lights Engine Turn Over Engine Start Case II LP * RP SP 2 73
Case Analysis Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Battery Fuel Line Starter Distributor Battery Power Gas Distributor Battery Power Spark Plugs Gas Gauge Radio Lights Engine Turn Over Gas Gauge Engine Start Radio Case I LP * RP Lights Engine Turn Over Engine Start Case II + LP * RP SP 2 74
Case Analysis Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Distributor Battery Power Spark Plugs Gauge Radio Lights LP * RP Engine Start Engine Turn Over + LP * RP SP 2 75
n n Decomposition and Case Analysis can answer any query Non-Deterministic! Battery Age Alternator Fan Belt Leak Charge Delivered Battery Fuel Line Starter Gas Battery Fuel Line Starter Distributor Battery Power Gas Distributor Battery Power Spark Plugs Gas Gauge Radio Lights Engine Turn Over Gas Gauge Engine Start Radio SP 2 Lights Engine Turn Over Engine Start 76
RC 1(T, e) if T is a leaf node return Lookup(T, e) else p : = 0 for each instantiation c of cutset(T)-E do p : = p + RC 1(Tl, ec) RC 1(Tr, ec) return p SP 2 77
Lookup(T, e) QX|U : CPT associated with leaf T If X is instantiated in e, then x: value of X in e u: value of U in e Return qx|u Else return 1 = Sx qx|u SP 2 78
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