Reasoning with Bayesian Networks
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1 Complexity of Probbilistic Inference Compiling Byesin Networks Resoning with Byesin Networks Lecture 5: Complexity of Probbilistic Inference, Compiling Byesin Networks Jinbo Hung NICTA nd ANU Jinbo Hung Resoning with Byesin Networks
2 Complexity of Probbilistic Inference Compiling Byesin Networks Decision Versions of Queries Complexity Clsses Hrdness nd Membership D-MAR / D-PR: Pr(q e) > p? D-MPE: In there x such tht Pr(x, e) > p? D-MAP: Given vribles Q X, in there q such tht Pr(q, e) > p? Jinbo Hung Resoning with Byesin Networks
3 Complexity of Probbilistic Inference Compiling Byesin Networks Decision Versions of Queries Complexity Clsses Hrdness nd Membership D-MAR / D-PR: Pr(q e) > p? D-MPE: In there x such tht Pr(x, e) > p? D-MAP: Given vribles Q X, in there q such tht Pr(q, e) > p? D-MPE is NP-complete D-MAR / D-PR is PP-complete D-MAP is NP PP -complete Jinbo Hung Resoning with Byesin Networks
4 Complexity of Probbilistic Inference Compiling Byesin Networks NP (Nondeterministic Polynomil) Complexity Clsses Hrdness nd Membership Solvble by nondeterministic Turing mchine in polynomil time Alterntive definition: YES nswers hve proofs tht cn be verified in polynomil time Problem Q is NP-complete if every problem in NP cn be reduced to Q in polynomil time Intuitively, hrdest problems in NP; no known (deterministic) polynomil time lgorithms Jinbo Hung Resoning with Byesin Networks
5 Complexity of Probbilistic Inference Compiling Byesin Networks SAT: An NP-complete Problem Complexity Clsses Hrdness nd Membership (X 1 X 2 X 3 ) ((X 3 X 4 ) X 5 ) Is there ssignment tht stisfies Boolen formul? SAT NP: Guess ssignment nd return YES if stisfying, in polynomil time NP-complete: Reduce computtion of nondeterministic Turing mchine to SAT Jinbo Hung Resoning with Byesin Networks
6 Complexity of Probbilistic Inference Compiling Byesin Networks PP (Probbilistic Polynomil) Complexity Clsses Hrdness nd Membership Solvble in polynomil time by nondeterministic Turing mchine tht ccepts when mjority (> hlf) of pths ccept Alterntive definition: Solvble in polynomil time with correctness probbility > 1/2 on YES instnces nd 1/2 on NO instnces NP PP SAT PP: Guess ssignment, if stisfying return YES, else return YES with probbility 1/2 Jinbo Hung Resoning with Byesin Networks
7 Complexity of Probbilistic Inference Compiling Byesin Networks MAJSAT: A PP-complete Problem Complexity Clsses Hrdness nd Membership (X 1 X 2 X 3 ) ((X 3 X 4 ) X 5 ) Are mjority of ssignments stisfying? MAJSAT PP: Why? PP-complete: Estblish one-to-one correspondence between computtion pths of nondeterministic Turing mchine nd stisfying ssignments of Boolen formul Jinbo Hung Resoning with Byesin Networks
8 NP PP Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Solvble in nondeterministic polynomil time given PP orcle E-MAJSAT (NP PP -complete): Is there ssignment to X 1,..., X k such tht mjority of its completions (to full ssignments) stisfy Boolen formul? Jinbo Hung Resoning with Byesin Networks
9 Complexity of Probbilistic Inference Compiling Byesin Networks Boolen Formul to Byesin Network Complexity Clsses Hrdness nd Membership (X 1 X 2 X 3 ) ((X 3 X 4 ) X 5 ) X1 X 2 X 3 X X 4 5 S Turn Boolen formul into Byesin network with deterministic CPTs (0/1 probbilities) Jinbo Hung Resoning with Byesin Networks
10 Complexity of Probbilistic Inference Compiling Byesin Networks Boolen Formul to Byesin Network Complexity Clsses Hrdness nd Membership (X 1 X 2 X 3 ) ((X 3 X 4 ) X 5 ) X1 X 2 X 3 X X 4 5 S Pr(x 1,..., x n, S α = true) = 1/2 n if x 1,..., x n stisfying, 0 otherwise Jinbo Hung Resoning with Byesin Networks
11 Complexity of Probbilistic Inference Compiling Byesin Networks Reducing SAT to D-MPE Complexity Clsses Hrdness nd Membership ssignment x 1,..., x n stisfying formul α iff vrible instntition y of network N α such tht Pr(y, S α = true) > 0 Jinbo Hung Resoning with Byesin Networks
12 Complexity of Probbilistic Inference Compiling Byesin Networks Reducing MAJSAT to D-PR/D-MAR Complexity Clsses Hrdness nd Membership Mjority of ssignments x 1,..., x n stisfy formul α iff Pr(S α = true) > 1/2 Jinbo Hung Resoning with Byesin Networks
13 Complexity of Probbilistic Inference Compiling Byesin Networks Reducing E-MAJSAT to D-MAP Complexity Clsses Hrdness nd Membership ssignment x 1,..., x k such tht mjority of ssignments x k+1,..., x n stisfy formul α iff MAP instntition x 1,..., x k such tht Pr(x 1,..., x k, S α = true) > 1/2 k+1 Jinbo Hung Resoning with Byesin Networks
14 Hrdness Summry Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership D-MPE is NP-hrd D-MAR / D-PR is PP-hrd D-MAP is NP PP -hrd Jinbo Hung Resoning with Byesin Networks
15 D-MPE NP Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership In there x such tht Pr(x, e) > p? Guess network instntition x (liner time) Compute Pr(x, e) (liner time, by chin rule) Return YES iff Pr(x, e) > p Jinbo Hung Resoning with Byesin Networks
16 Complexity of Probbilistic Inference Compiling Byesin Networks D-MAR/D-PR PP Complexity Clsses Hrdness nd Membership Pr(q e) > p? Smple network instntition x (liner time) Instntite vribles X one t time, prents before child Smple vlue ccording to probbilities in CPT Return YES with probbility (p) = min(1, 1/2p) if x eq b(p) = mx(0, (1 2p)/(2 2p)) if x e, x q 1/2 if x e Pr(YES) > 1/2 exctly when Pr(q e) > p Jinbo Hung Resoning with Byesin Networks
17 D-MAP NP PP Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Given vribles Q X, in there q such tht Pr(q, e) > p? Guess instntition q (liner time) Check if Pr(q, e) > p ( D-PR problem) using PP orcle Return YES iff Pr(q, e) > p Jinbo Hung Resoning with Byesin Networks
18 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Summry Complexity Clsses Hrdness nd Membership D-MPE is NP-complete D-MAR / D-PR is PP-complete D-MAP is NP PP -complete Jinbo Hung Resoning with Byesin Networks
19 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership D-MAP on Polytrees (with bounded # of prents / node) Given vribles Q X, in there q such tht Pr(q, e) > p? NP Guess instntition q (liner time) Check if Pr(q, e) > p by vrible elimintion (polynomil time due to bounded treewidth) Return YES iff Pr(q, e) > p NP-hrd by reduction from MAXSAT Jinbo Hung Resoning with Byesin Networks
20 Complexity of Probbilistic Inference Compiling Byesin Networks MAXSAT (decision version) Complexity Clsses Hrdness nd Membership Given cluses α 1,..., α m over vribles X 1,..., X n, is there ssignment x 1,..., x n stisfying more thn k cluses? NP: Guess ssignment, check # of stisfied cluses, return YES if > k NP-hrd: Contins SAT s specil cse (k = m 1) Jinbo Hung Resoning with Byesin Networks
21 Complexity of Probbilistic Inference Compiling Byesin Networks Reducing MAXSAT to D-MAP Complexity Clsses Hrdness nd Membership X 1 X 2 X 3 X n S 0 S 1 S 2 S 3 S n S 0 : {1,..., m}, uniform priors on S 0, X 1,..., X n For j > 0, S j = 0 if one of x 1,..., x j stisfies α i, where i = S 0 ; S j = i otherwise Jinbo Hung Resoning with Byesin Networks
22 Complexity of Probbilistic Inference Compiling Byesin Networks Reducing MAXSAT to D-MAP Complexity Clsses Hrdness nd Membership X 1 X 2 X 3 X n S 0 S 1 S 2 S 3 S n ssignment x 1,..., x n stisfying more thn k cluses iff instntition x 1,..., x n such tht Pr(x 1,..., x n, S n = 0) > k/(m2 n ) Jinbo Hung Resoning with Byesin Networks
23 Complexity of Probbilistic Inference Compiling Byesin Networks Weighted Model Counting Complexity Clsses Hrdness nd Membership of weights of models of Boolen formul Weight for literl: A (.3), A (.7), B (1), B (1) Weight of model: w(ab) =.7 1 =.7 Weight of formul: w(a B) = w(ab) + w(ab) + w(ab) = = 1.3 Jinbo Hung Resoning with Byesin Networks
24 Complexity of Probbilistic Inference Compiling Byesin Networks Weighted Model Counting Complexity Clsses Hrdness nd Membership of weights of models of Boolen formul Weight for literl: A (.3), A (.7), B (1), B (1) Weight of model: w(ab) =.7 1 =.7 Weight of formul: w(a B) = w(ab) + w(ab) + w(ab) = = 1.3 Use weighted model counters to compute Pr(e) Encode Byesin network into Boolen formul with weights Jinbo Hung Resoning with Byesin Networks
25 First Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Indictor vribles/cluses I 1 I 2 I b1 I b2 I c1 I c2 I 1 I 2 I b1 I b2 I c1 I c2 Jinbo Hung Resoning with Byesin Networks
26 First Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Prmeter vribles/cluses I 1 P 1 I 2 P 2 I 1 I b1 P b1 1 I 1 I b2 P b2 1 I 2 I b1 P b1 2 I 2 I b2 P b2 2 Jinbo Hung I 1 I c1 P c1 1 I 1 I c2 P c2 1 I 2 I c1 P c1 2 I 2 I c2 P c2 2 Resoning with Byesin Networks
27 First Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Given e = e 1,..., e k Pr(e) = w( N I e1... I ek ) where w(p x u ) = x u, w(i x ) = w( I x ) = w( P x u ) = 1 Jinbo Hung Resoning with Byesin Networks
28 First Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Pr( 1 c 2 ) = w( N I 1 I c2 ) = w(ω 1 ) + w(ω 3 ) = =.09 Jinbo Hung Resoning with Byesin Networks
29 Second Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Indictor vribles/cluses (s 1st encoding) I 1 I 2 I 3 I 1 I 2 I 1 I 3 I 2 I 3 I b1 I b2 I b1 I b2 Jinbo Hung Resoning with Byesin Networks
30 Second Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Prmeter vribles/cluses (fewer thn 1st encoding) Q 1 I 1 Q 1 Q 2 I 2 Q 1 Q 2 I 3 I 1 Q b1 1 I b1 I 1 Q b1 1 I b2 I 2 Q b1 2 I b1 I 2 Q b1 2 I b2 I 3 Q b1 3 I b1 I 3 Q b1 3 I b2 Jinbo Hung Resoning with Byesin Networks
31 Second Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Multiple models for sme instntition of I 1, I 2, I 3, I b1, I b2 Eight Models for I 1 I 2 I 3 I b1 I b2 Q 1 Q 2 Q b1 2 Q 3, Q b1 1, Q b1 3 unconstrined Given pproprite weights, Pr(x) = model x w(model) Jinbo Hung Resoning with Byesin Networks
32 Second Encoding Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Given e = e 1,..., e k Pr(e) = w( N I e1... I ek ) where w(q x u ) = x u 1 x <x x u, w( Q x u ) = 1 w(q x u ), w(i x ) = w( I x ) = 1 Jinbo Hung Resoning with Byesin Networks
33 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership (X Y Z) 3, ( X ) 10.1, (Y ).5, (Z) 2.5 Cluses hve weights W-MAXSAT: Find ssignment mximizing sum of weights of stisfied cluses Alterntively: Minimize penlty sum of weights of violted cluses Jinbo Hung Resoning with Byesin Networks
34 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Indictor cluses, ll hving weight W (some big number) I 1 I 2 I 3 I 1 I 2 I 1 I 3 I 2 I 3 I b1 I b2 I b1 I b2 These hrd cluses ensure ssignment corresponds to network instntition Jinbo Hung Resoning with Byesin Networks
35 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Prmeter cluses log.3 ( I 1 ) log.5 ( I 2 ) log.2 ( I 3 ) log.2 ( I 1 I b1 ) log.8 ( I 1 I b2 ) ( I 2 I b1 ) log 1 ( I 2 I b2 ) W log.6 ( I 3 I b1 ) log.4 ( I 3 I b2 ) Jinbo Hung Resoning with Byesin Networks
36 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership I 1 I 2 I 3 I b1 I b2 hs penlty log.3 log.8 = log(.3.8) = Pr( 1, b 2 ) Prmeter cluses log.3 ( I 1 ) log.5 ( I 2 ) log.2 ( I 3 ) log.2 ( I 1 I b1 ) log.8 ( I 1 I b2 ) ( I 2 I b1 ) log 1 ( I 2 I b2 ) W log.6 ( I 3 I b1 ) log.4 ( I 3 I b2 ) Jinbo Hung Resoning with Byesin Networks
37 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership I 1 I 2 I 3 I b1 I b2 hs penlty log.3 log.8 = log(.3.8) = Pr( 1, b 2 ) Any ssignment Γ (stisfying hrd cluses) violtes exctly one cluse from ech CPT Pn(Γ) = log x u x u x = log x u x x u = log Pr(x) Min penlty mx Pr(x) Jinbo Hung Resoning with Byesin Networks
38 Complexity of Probbilistic Inference Compiling Byesin Networks Complexity Clsses Hrdness nd Membership Complexity of Probbilistic Inference: Summry D-MPE NP-complete, D-MAR / D-PR PP-complete, D-MAP NP PP -complete D-MAP on polytrees NP-complete Two encodings for reducing Pr(e) to weighted model counting Reducing MPE to weighted MAXSAT Jinbo Hung Resoning with Byesin Networks
39 Arithmetic Circuits Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Jinbo Hung Resoning with Byesin Networks
40 Arithmetic Circuits Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion + λ λ b λb b b λ b b c λc c c λc c Jinbo Hung Resoning with Byesin Networks
41 Arithmetic Circuits Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion.1 + λ λ b λb b b λ b b c λc c c λc c Jinbo Hung Resoning with Byesin Networks
42 Complexity of Probbilistic Inference Compiling Byesin Networks Network Polynomil Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion A B Pr(A, B) b b b b b b b b Jinbo Hung Resoning with Byesin Networks
43 Complexity of Probbilistic Inference Compiling Byesin Networks Network Polynomil Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion A B Pr(A, B) b λ λ b b b λ λ b b b λ λ b b b λ λ b b Jinbo Hung Resoning with Byesin Networks
44 Complexity of Probbilistic Inference Compiling Byesin Networks Network Polynomil Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion A B Pr(A, B) b λ λ b b b λ λ b b b λ λ b b b λ λ b b f = λ λ b b + λ λ b b + λ λ b b + λ λ b b Jinbo Hung Resoning with Byesin Networks
45 Complexity of Probbilistic Inference Compiling Byesin Networks Network Polynomil Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion A B Pr(A, B) b λ λ b b b λ λ b b b λ λ b b b λ λ b b f = λ λ b b + λ λ b b + λ λ b b + λ λ b b f (e = ) = (0)(1) b (0)(1) b + (1)(1) b + (1)(1) b = b + b = Pr(e) Jinbo Hung Resoning with Byesin Networks
46 Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Network Polynomil s Arithmetic Circuit f = λ λ b λ c b c + λ λ b λ c b c λ λ b λ c b c + λ λ b λb b b λ b b c λc c c λc c Jinbo Hung Resoning with Byesin Networks
47 Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Network Polynomil s Arithmetic Circuit f = λ λ b λ c b c + λ λ b λ c b c λ λ b λ c b c + λ λ b λb b b λ b b c λc c c λc c f hs exponentil size, circuit my not Jinbo Hung Resoning with Byesin Networks
48 Prtil Derivtives Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Pr(e = c) = λ λ b λb b b λ b b c λc c c λc c Jinbo Hung Resoning with Byesin Networks
49 Prtil Derivtives Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Pr(e = c) = f is liner in every vrible f λ (e) =.4 λ b λb b b λ b b c λc c c λc c λ f will increse by.4 if λ chnges from 0 to 1 (e chnges from c to c) Pr(c) = =.5 Jinbo Hung Resoning with Byesin Networks
50 Prtil Derivtives Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Pr(e = c) = f λ x (e) = Pr(x, e X ) Flipping vrible: x e λ b λb b b λ b b c λc c c λc c λ Adding literl: x, x e Pr(e): x e Pr( c) = f λ =.4 Pr(bc) = f λ b =.1 Jinbo Hung Resoning with Byesin Networks
51 Prtil Derivtives Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Pr(e = c) = x u f x u (e) = Pr(x, u, e) Gives fmily mrginls Pr(x, u, e) xu λ b λb b b λ b b c λc c c λc c λ f x u lso useful in sensitivity nlysis nd prmeter lerning Jinbo Hung Resoning with Byesin Networks
52 Complexity of Probbilistic Inference Compiling Byesin Networks Evlution nd Differentition Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ λ b 1.1 λb 1.1 b b λ b 1 0 b c λc c c λc c Jinbo Hung Resoning with Byesin Networks
53 Complexity of Probbilistic Inference Compiling Byesin Networks Evlution nd Differentition Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Bottom-up pss evlutes circuit, computes Pr(e) Top-down pss computes ll prtil derivtives Liner in circuit size Use division insted of multipliction Hndle 0s with extr bit per node, set to 1 when exctly one child hs 0 vlue Jinbo Hung Resoning with Byesin Networks
54 Mximizer Circuits Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Turn ddition into mximiztion in both psses Bottom-up pss computes f m = MPE P (e) Cn recover MPE instntition Top-down pss computes ll prtil derivtives f m x λ x (e) = MPE P (x, e X ) f m x,u x u x u (e) = MPE P (e, x, u) f m x (f m x,u) is mx over terms x (xu) Jinbo Hung Resoning with Byesin Networks
55 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ A B Θ B A true true n 3 = (λ b, b ) true flse n 4 = (λ b, b ) flse true n 5 = (λ b, b ) flse flse n 6 = (λ b, b ) A Θ A true n 1 = (λ, ) flse n 2 = (λ, ) Jinbo Hung Resoning with Byesin Networks
56 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ A B Θ B A true true n 3 = (λ b, b ) true flse n 4 = (λ b, b ) flse true n 5 = (λ b, b ) flse flse n 6 = (λ b, b ) Jinbo Hung Resoning with Byesin Networks
57 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ A B Θ B A true true n 3 = (λ b, b ) true flse n 4 = (λ b, b ) flse true n 5 = (λ b, b ) flse flse n 6 = (λ b, b ) A B Θ B A true n 7 = +(n 3, n 4 ) flse n 8 = +(n 5, n 6 ) Jinbo Hung Resoning with Byesin Networks
58 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ A Θ A B Θ B A true n 9 = (n 1, n 7 ) flse n 10 = (n 2, n 8 ) Jinbo Hung Resoning with Byesin Networks
59 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ A Θ A B Θ B A true n 9 = (n 1, n 7 ) flse n 10 = (n 2, n 8 ) A Θ A B Θ B A n 11 = +(n 9, n 10 ) Jinbo Hung Resoning with Byesin Networks
60 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Vrible Elimintion 11 + Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion λ b λb b b λb b λ Circuit size O(n exp(w)) s complexity of vrible elimintion Jinbo Hung Resoning with Byesin Networks
61 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits Embedded in Jointree Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion + λ A A A root λ λ A A AB AC b λb b b λ b b c λc c c λc c λ B B A λ C C A Cluster instntition to node, seprtor instntition to + node Jinbo Hung Resoning with Byesin Networks
62 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits Embedded in Jointree Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion + λ A A A root λ λ A A AB AC b λb b b λ b b c λc c c λc c λ B B A λ C C A Circuit size O(n exp(w)) s complexity of jointree propgtion Jinbo Hung Resoning with Byesin Networks
63 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Boolen Encoding Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion I I I b I b I I I b I b I P I P I I b P b I I b P b I I b P b I I b P b Jinbo Hung Resoning with Byesin Networks
64 Complexity of Probbilistic Inference Compiling Byesin Networks Circuits from Boolen Encoding Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Boolen encoding (1st version) of Byesin network encodes network polynomil Ech model corresponds to one term Convert Boolen formul to Boolen circuit stisfying four properties NNF: literls s leves,, s internl nodes Decomposble: children of node do not shre vribles Determinism: children of node mutully inconsistent Smooth: children of node hve sme vribles Conversion lgorithms studied in Knowledge Compiltion Convert Boolen circuit to rithmetic circuit Jinbo Hung Resoning with Byesin Networks
65 Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Boolen Circuit to Arithmetic Circuit I P b I P P I b I I P P I P b b P b I b I P b b λ λ P b P P b P b b b λb b b λb b to +, to, negtive literls to 1 I x to λ x, P x u to x u Jinbo Hung Resoning with Byesin Networks
66 Complexity of Probbilistic Inference Compiling Byesin Networks Arithmetic Circuits Differentil Semntics Circuit Propgtion Circuit Compiltion Compiling Byesin Networks: Summry Network polynomils nd rithmetic circuits Differentil semntics of rithmetic circuits Circuit evlution nd differentition, mximizer circuits Circuits compiled from vrible elimintion, jointree, Boolen encoding Jinbo Hung Resoning with Byesin Networks
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