Probabilistic Soft Logic for Personalized Medicine
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1. The document describes Probabilistic Soft Logic (PSL), a probabilistic modeling language based on logics. 2. PSL uses rules to capture dependencies and constraints between continuous random variables represented as atoms. A PSL program consists of rules, sets, constraints, and atoms. 3. PSL provides a mathematical foundation based on constrained continuous Markov random fields and a logical foundation based on generalized annotated logic programs. It allows for collective probabilistic inference and learning over relational domains.
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Probabilistic Soft Logic for Personalized Medicine
- 1. Probabilistic Soft Logic by woodleywonderworks © Probabilistic Soft Logic Matthias Bröcheler Lilyana Mihalkova, Stephen Bach, Stanley Kok, and Lise Getoor
- 2. Probabilistic Soft Logic Applications 1. Ontology Alignment 2. Personalized Medicine 2 3. Diffusion Modeling
- 3. Probabilistic Soft Logic Ontologies provides work for Organization buys interacts Service & Products Customers Employees develops sells to helps Software Hardware IT Services Developer Sales Person Staff Networks Process Optim. ERP Systems Accountant = sub-concept Instance data not shown! relationship 3
- 4. Probabilistic Soft Logic Ontologies provides work for Organization buys interacts Service & Products Customers Employees develops sells to helps Software Hardware IT Services Developer Sales Person Database Schema Staff Networks Process Optim. ERP Systems Accountant = sub-concept Instances not shown! relationship 4
- 5. Probabilistic Soft Logic Multiple Ontologies provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 5
- 6. Probabilistic Soft Logic Ontology Alignment [3 ] provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 6
- 7. Probabilistic Soft Logic Ontology Alignment provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff Match, Don’t Match? develop Company works for buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 7
- 8. Probabilistic Soft Logic Ontology Alignment provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff Similar to what extent? develop Company works for buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 8
- 9. Probabilistic Soft Logic Personalized Medicine [2 ] Joe Black Age: 51 BMI: 27 Diet: high in fat Rectal exam: no signs PSA (blood test): 5.2 Mutations on: LMTK2, KLK3, JAZF1 Discomfort when urinating Example Diagnosis and Treatment of Prostate Cancer
- 10. Probabilistic Soft Logic Bob Black Joe Black Died at age 79 Age: 51 Never diagnosed with father BMI: 27 prostate cancer Diet: high in fat PSA levels: 3.2-8.9 Rectal exam: no signs BMI: 23 PSA (blood test): 5.2 Mutations on: LMTK2, KLK3, JAZF1 Frank Black Discomfort when urinating Age: 48 BMI: 24 PSA: 3.1, 4.2, 4.9, 55 Mary Black Biopsy: 8/12 positive brother wife Age: 45 Grade P1: 2-3, 60/40 BMI: 32 Grade P2: 4-5, 90/10 Diet: high in fat Mutations on: Diagnosed with LMTK2, KLK3, JAZF1, breast cancer, CDH13 XMRV virus detected
- 11. Probabilistic Soft Logic Bob Black Joe Black Died at age 79 Age: 51 Never diagnosed with father BMI: 27 prostate cancer Diet: high in fat PSA levels: 3.2-8.9 Rectal exam: no signs BMI: 23 PSA (blood test): 5.2 Mutations on: LMTK2, KLK3, JAZF1 Frank Black Discomfort when urinating Age: 48 BMI: 24 PSA: 3.1, 4.2, 4.9, 55 Support Medical Mary Black Biopsy: 8/12 positive Grade P1: 2-3, 60/40 Decision Making brother wife Age: 45 BMI: 32 Grade P2: 4-5, 90/10 Diet: high in fat Mutations on: Diagnosed with LMTK2, KLK3, JAZF1, breast cancer, CDH13 XMRV virus detected
- 12. Probabilistic Soft Logic Diffusion in Social Networks [4 ] Diffusion is a widely studied dynamic of social networks - Epidemiology • SIR Disease Model - Marketing • Viral Marketing - Health • Obesity Study - Campaign Management © Christakis, Fowler • Opinion Leaders 12
- 13. Probabilistic Soft Logic 500 million users 50M tweets / day Data is available © Ludwig Gatzke
- 14. Probabilistic Soft Logic Voter Opinion Modeling ?
- 15. Probabilistic Soft Logic Voter Opinion Modeling ? Status update $ $ Tweet
- 16. Probabilistic Soft Logic Voter Opinion Modeling spouse colleague friend friend spouse friend friend spouse colleague
- 17. Probabilistic Soft Logic What’s the commonality? Collective Probabilistic Reasoning in Relational Domains 17
- 18. Probabilistic Soft Logic What’s the commonality? Collective Probabilistic Reasoning in Relational Domains Statistical Relational Learning [Getoor & Taskar ’07] 18
- 19. Probabilistic Soft Logic SRL Alphabet Soup 19
- 20. Probabilistic Soft Logic SRL Alphabet Soup PSL? 20
- 21. Probabilistic Soft Logic Why PSL? Continuous Random Variables Mathematical Foundation Logic Foundation Inference & Learning Sets and Aggregators Extensible High Performance 21
- 22. Probabilistic Soft Logic What is PSL? Declarative language based on logics to express collective probabilistic inference problems - Predicate = relationship or property - (Ground) Atom = (continuous) random variable - Rule = capture dependency or constraint - Set = define aggregates PSL Program = Rules, Sets, Constraints, Atoms 22
- 23. Probabilistic Soft Logic Ontology Alignment similar(A,B) [A≈B] provides Organization work for buys interacts Service & Products similar(Customer,Customers) Customers Employees develops helps sells to [Customer≈Customers] Software Hardware IT Services Developer Sales Person Staff domain(C,D) develop works for Company domain(work for, Employees) buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 23
- 24. Probabilistic Soft Logic Ontology Alignment provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products & Services Customer Employee sells helps R≈T Software Dev Hardware ôConsulting Technician domainOf(R,A) domainOf(T,B) Sales Accountant 24 A≈B R≠T : 0.7
- 25. Probabilistic Soft Logic {A.subConcept}≈{B.subConcept} ô A≠B Ontology Alignment A≈B type(A,concept) type(B,concept) :0.8 provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 25
- 26. Probabilistic Soft Logic Ontology Alignment provides work for Organization buys interacts Service & Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products & Services Customer Employee sells helps Software Dev Hardware similar := partial-functional Accountant Consulting Technician Sales 26 := inverse partial-functional
- 27. Probabilistic Soft Logic Voter Opinion Modeling vote(A,P) friend(B,A) vote(B,P) : 0.3 spouse colleague friend friend spouse friend friend spouse colleague vote(A,P) spouse(B,A) vote(B,P) : 0.8
- 28. Probabilistic Soft Logic Mathematical Foundation
- 29. Probabilistic Soft Logic CCMRF Constrained Continuous Markov Random Field Markov Random Field - Undirected - Entropy-maximizing Continuous (Random Variables) Constrained (Domain) 29
- 30. Probabilistic Soft Logic CMRF RVs Range of RVs Domain of MRF n X = {X1 , .., Xn } : Di ⊂R D= ×i=1 Di Feature or Compatibility Kernels Parameters φ = {φ1 , .., φm } : φj : D → [0, M] ; Λ = {λ1 , .., λm } Probability measure P over X defined through m Density 1 f (x) = exp[− λj φj (x)] Function Z(Λ) j=1 m Partition Z(Λ) = exp − λj φj (x) dx Function D j=1 30
- 31. Probabilistic Soft Logic CCMRF : Constraints [5 ] Equality Constraints kA kA A(x) = a where A : D → R ,a ∈ R Inequality Constraints kB kB B(x) ≤ b where B : D → R ,b ∈ R Restricted Domain ˜ D = D ∩ {x|A(x) = a ∧ B(x) ≤ b} Adjusted CCMRF / ˜ f (x) = 0 ∀x ∈ D 31
- 32. Probabilistic Soft Logic Geometric Intuition X1 1 x1 + x3 ≤ 1 φ1 (x) = x1 φ2 (x) = max(0, x1 − x2 ) φ3 (x) = max(0, x2 − x3 ) X3 0 1 Λ = {1, 2, 1} X2 X = {X1 , X2 , X3 } 32
- 33. Probabilistic Soft Logic Geometric Intuition X1 1 x1 + x3 ≤ 1 φ1 (x) = x1 φ2 (x) = max(0, x1 − x2 ) φ3 (x) = max(0, x2 − x3 ) Highest Probability X3 0 1 Λ = {1, 2, 1} X2 X = {X1 , X2 , X3 } 33
- 34. Probabilistic Soft Logic Logic Foundation - Syntax Semantics -
- 35. Probabilistic Soft Logic Rules Ground Atoms [3 ] H1. ... Hm ô B1 , B2 ,... Bn h Atoms are real valued - Interpretation I, atom A: I(A) [0,1] - We will omit the interpretation and write A [0,1] h is a combination function - Arbitrary T-norms: [0,1]n Ø [0,1] Based on the theory of Generalized Annotated Logic Programs (GAP) [Kifer Subrahmanian ‘92] - But restricted to real values 35
- 36. Probabilistic Soft Logic Rules H1. ... Hm ô B1 , B2 ,... Bn h h is a combination function - Lukasiewicz T-norm ⊕ (h1, h2) = min(1, h1+h2 ) ⊗ (h1, h2) = max(0, 1- h1+h2 ) We use the Lukasiewicz T-norm in the following. 36
- 37. Probabilistic Soft Logic Satisfaction H1. ... Hm ô B1 , B2 ,... Bn Establish Satisfaction - ⊕(H1,..,Hm) ¥ ⊗(B1,..,Bn) R≈T:?ôA≈B:0.7 D≈E:0.8 Interpretation implicit! 37
- 38. Probabilistic Soft Logic Satisfaction H1. ... Hm ô B1 , B2 ,... Bn Establish Satisfaction - ⊕(H1,..,Hm) ¥ ⊗(B1,..,Bn) R≈T:≥0.5ôA≈B:0.7 D≈E:0.8 Interpretation implicit! 38
- 39. Probabilistic Soft Logic Distance to Satisfaction H1. ... Hm ô B1 , B2 ,... Bn Distance to Satisfaction - max( ⊗(B1,..,Bn) - ⊕(H1,..,Hm) , 0) R≈T:0.7ôA≈B:0.7 D≈E:0.8 0.0 R≈T:0.2ôA≈B:0.7 D≈E:0.8 0.3 39
- 40. Probabilistic Soft Logic Rule Weights R: H1. ... Hm ô B1 , B2 ,... Bn w Weighted Distance to Satisfaction - d(R,I) = w * max(⊗ (B1,..,Bn)- ⊕ (H1,..,Hm), 0) 40
- 41. Probabilistic Soft Logic Rule Weights R: H1. ... Hm ô B1 , B2 ,... Bn w Weighted Distance to Satisfaction - d(R,I) = w * max(⊗ (B1,..,Bn)- ⊕ (H1,..,Hm), 0) Every ground rule R in a PSL program P contributes a compatibility kernel ϕR = d(R,I) to the CCMRF associated with P. 41
- 42. Probabilistic Soft Logic Geometric Intuition R2 1 | d(R2,I) | 0 | | d(R1,I) 1 R1 42
- 43. Probabilistic Soft Logic Geometric Intuition 1 R2 P(I|P ) = exp [−d(P, I)] Z(w) w2 | d(R2) | norm = d(P,I) 0 | | d(R1) w1 R1 43
- 44. Probabilistic Soft Logic Geometric Intuition X1 1 x1 + x3 ≤ 1 φ1 (x) = x1 φ2 (x) = max(0, x1 − x2 ) φ3 (x) = max(0, x2 − x3 ) Highest Probability X3 0 1 Λ = {1, 2, 1} X2 X = {X1 , X2 , X3 } 44
- 45. Probabilistic Soft Logic Inference - MAP Marginals -
- 46. Probabilistic Soft Logic MAP Inference [3 ] Most Probable Interpretation - Most likely truth value assignment given some facts. argmax ( I | P) I ñ argmin d(P,I) I 46
- 47. Probabilistic Soft Logic MAP Inference Theory Exact PSL inference in polynomial time - Convex optimization problem due to our choices in combination functions O(n3.5) inference - Second Order Cone Program - n=number of (active) ground rules - Efficient commercial optimization packages 47
- 48. Probabilistic Soft Logic Inference Algorithm Each ground rule constitutes a linear or conic constraint introducing a rule specific “dissatisfaction” variable that is added to the objective function. 48
- 49. Probabilistic Soft Logic Inference Algorithm Conservative Grounding: Most rules trivially have satisfaction distance=0. Save time and space by not grounding them out in the first place. Don’t reason about it if you don’t absolutely have to! 49
- 50. Probabilistic Soft Logic Parallelizing MAP Inference [4 ] MAP inference is O(n3.5) - Limited scalability Achieve scalability by dividing inference problem into smaller “chunks” - Allows for parallelization and distribution of workload - Similar to message-passing but on entire subgraphs of the factor graph 50
- 51. Probabilistic Soft Logic Factor Graph vote(Mary,Dem) vote(Jane, Dem) vote(Mary,Dem) spouse(John,Mary) vote(Jane,Dem) friend(John,Jane) vote(John,Dem) : 0.8 vote(John,Dem) : 0.3 vote(John,Dem) 51
- 52. Probabilistic Soft Logic Factor Graph vote(Mary,Dem) vote(Jane, Dem) vote(Mary,Dem) spouse(John,Mary) vote(Jane,Dem) friend(John,Jane) vote(John,Dem) : 0.8 vote(John,Dem) : 0.3 vote(John,Dem) Idea: Partition Dependency graph into strongly connected components and solve MAP on each independently 52
- 53. Probabilistic Soft Logic Approximate Algorithm 1. Ground out factor graph conservatively 2. Partition dependency graph using a modularity maximizing clustering alg - Inspired by Blondel et al [06] - Aggregate rule weights 3. Compute MAP on each cluster fixing confidence values of outside atoms 4. Go to 1 until change in I Θ 53
- 54. USA dean author Probabilistic Soft Logic member Prof Prof Jones Baneri Italy in Paper “ABC” comment author UC CS UMD CS in faculty friends faculty Prof Calero department in member faculty presented Prof Dooley attended Social Science department University MD Universita Calabria department in dean ASONAM 09 attended faculty submitted Prof Roma author UMD Physics author member visited organized accepted friends author KPLLC Paper 09 “UVW” S3 Prof Smith Paper “HIJ” submitted Paper “XYZ” comment attended comment student of S2 student of Prof Olsen collaborates Prof Lund member dean Prof Larsen faculty Jamie Lock member Karl Oede Social Science visited Odense SDU Physics Odense colleagues John Doe department Denmark
- 55. Probabilistic Soft Logic Scalability 16000 14000 Exact vs Approximate Algorithm Running Times 12000 Time in Seconds Exact Algorithm 10000 Approximate Algorithm with Parameters A 8000 6000 4000 2000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 # Compatibility Kernels in Graph 55
- 56. Probabilistic Soft Logic Accuracy 7% Relative Error compared to Exact Inference 6% Percentage Relative Error 5% Parameters B 4% Parameters A 3% Parameters C Parameters D 2% Parameters E 1% 0% 0 10000 20000 30000 40000 50000 60000 70000 80000 Number of Compatibility Kernels 56
- 57. Probabilistic Soft Logic Runtime Running Time Comparison of Approximate 500 Algorithm 450 Parameters B 400 Parameters A 350 Parameters C Time in Seconds 300 Parameters D 250 Parameters E 200 150 100 50 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Number of Compatibility Kernels 57
- 58. Probabilistic Soft Logic Accuracy on very large Graphs Relative Error Comparison 6% 5% Percentage Relative Error Parameters B Parameters C 4% Parameters D 3% Parameters E 2% 1% 0% 3.5E+05 7.0E+05 1.4E+06 2.8E+06 5.6E+06 Number of Compatibility Kernels Log-scale 58
- 59. Probabilistic Soft Logic Runtime on very large Graphs Runtime Comparison 40000 Time in Seconds 4000 Parameters B Parameters A 2M edges Parameters C in 48 min Parameters D Parameters E 400 3.5E+05 7.0E+05 1.4E+06 2.8E+06 5.6E+06 Number of Compatibility Kernels Log-log-scale 59
- 60. Probabilistic Soft Logic Computing Marginals [5 ] For a subset of RVs X ⊂ X RV = atom - In our case X = {Xi } Compute the marginal density function fX (x ) = f (x , y)dy ˜ y∈×Di ,s.t.Xi ∈X / f | | 0 1 Technician≈Developer 60
- 61. Probabilistic Soft Logic Geometric Intuition X1 f 1 | | 0 1 P(0.4 ≤ X2 ≤ 0.6) X3 0 1 X2 61
- 62. Probabilistic Soft Logic Computing Marginal in Theory Computing the marginal probability density function for a subset X ⊂ X under the probability measure defined by a CCMRF is #P hard in the worst case. - Related to volume computation of polytopes, based on [Broecheler et al, ‘09] 62
- 63. Probabilistic Soft Logic Sampling Scheme Approximate the marginal distributions using an MCMC sampling scheme restricted to the convex polytope defined by D˜ - Again, inspired by work on volume computation 63
- 64. Probabilistic Soft Logic Histogram Sampling Xi 64
- 65. Probabilistic Soft Logic Histogram Sampling Xi 65
- 66. Probabilistic Soft Logic Histogram Sampling Xi 66
- 67. Probabilistic Soft Logic Histogram Sampling Xi 67
- 68. Probabilistic Soft Logic Random Ball Walk Need to sample from restricted to the ball difficult q1 r p q2 68
- 69. Probabilistic Soft Logic Hit-and-Run d p 69
- 70. Probabilistic Soft Logic Hit-and-Run q d Compute density function induced on p line and sample from it easy 70
- 71. Probabilistic Soft Logic Hit-and-Run q d p 71
- 72. Probabilistic Soft Logic Sampling in theory Theorem: The complexity of computing an approximate distribution σ* using the hit-and-run sampling scheme such that the total variation distance of σ* and P is less than ε is ∗ 3 O n (kB + n + m) ˜ ˜ where n = n − kA , under the assumptions that ˜ we start from an initial distribution σ such that the density function dσ/dP is bounded by M except on a set S with σ(S)≤ε/s [Lovasz Vempala ‘04] 72
- 73. Probabilistic Soft Logic Sampling in Practice Starting distribution = MAP state How do we get out of corners? 73
- 74. Probabilistic Soft Logic Sampling in Practice How do we get out of corners? Use relaxation method [Agmom ‘54] to solve system of linear inequalities to find a feasible direction d zk − W k d i T di+1 = di + 2 Wk ε1 Wk 2 ε2 74
- 75. Probabilistic Soft Logic Algorithm Convergence KL Divergence by Sample Size 5 KL Divergence 0.5 Average KL Divergence Lowest Quartile KL Divergence Highest Quartile KL Divergence 0.05 30000 300000 3000000 Number of Samples Averaged over 30 randomly snow-ball sampled folds Lowest Quartile = 322-413 atoms Highest Quartile = 174-224 atoms 75
- 76. Probabilistic Soft Logic Algorithm Performance Runtime for 1000 Samples 35 30 25 Time in sec 20 15 10 5 0 0 2000 4000 6000 8000 10000 Number of Compatibility Kernels 76
- 77. Probabilistic Soft Logic Weight Learning
- 78. Probabilistic Soft Logic Weight Learning [3 ] Given: Rules + Training Instance | Want: Weights w∗ = argmaxw P(IT |P ) − ||w||2 Approach: Maximize likelihood of observation by optimizing weights - Plus prior on weights Problem: Cannot compute partition function Z tractably Workaround: Use MAP state to approximate Z - Invoke reasoner during gradient computation - BFGS and Perceptron implementations 78
- 79. Probabilistic Soft Logic Similarity Reasoning
- 80. Probabilistic Soft Logic Experiments: Ontology Alignment OAEI Ontology Alignment Benchmark (2008) - Real world ontologies (300s) - Synthetic ontology pairs - Approx 100 entities - 21 rules, modified standard string similarity measures 80
- 81. Probabilistic Soft Logic OAEI comparison [3 ] 1 0.8 F1 Score 0.6 0.4 0.2 0 Other results as reported by the benchmark participants. 81
- 82. Probabilistic Soft Logic Attribute Similarity Functions A≈B ô A.name ≈x B.name Maximum flexibility for attribute similarity Customization to particular problem domains - Camel-case common in web-ontologies Users can define arbitrary similarity functions ≈x to be integrated into PSL - e.g. String similarity measures such as Levenshtein 82
- 83. Probabilistic Soft Logic Sets in PSL {A.subConcept}≈{B.subConcept} ô A≠B A≈B type(A,concept) type(B,concept) :0.8 provides Organization work for buys interacts Service Products Customers Employees develops helps sells to Software Hardware IT Services Developer Sales Person Staff develop works for Company buys interacts with Products Services Customer Employee sells helps Software Dev Hardware Consulting Technician Sales Accountant 83
- 84. Probabilistic Soft Logic Explicit Set Treatment A≈B ô {A.subConcept} ≈{} {B.subConcept} Reason about the similarity of sets of entities Allow to integrate aggregates measures Default Set equality measure: Jaccard-type 2 x∈X y∈Y x≈y X≈Y = |X| + |Y | - Allow users to define alternative set equalities • Based on inference engine • Initially, PSL provides some predefined set overlap measures 84
- 85. Probabilistic Soft Logic Support for Sets Using relational syntax… - X.name, X.father, X.friend (a friend) - Binary predicates only …makes it easier to specify sets - {X.friend} - all friends - {X.friend.friend} - all second level friends Inverse of binary relation - X.knows(inv) (who knows X?) Union, Intersection - {X.knows} u {X.knows(inv)} = {Y.knows} u {Y.knows(inv)} 85
- 86. Probabilistic Soft Logic Utility of Sets in PSL Compare set vs non-set version of rules on synthetic ontology alignment benchmark A≈B ô {A.subConcept} ≈{} {B.subConcept} vs A≈B ô A.subConcept ≈ B.subConcept 86
- 87. Probabilistic Soft Logic Ontology Set Comparison 1 Structural 0.9 Noise: 0.2 0.8 F1 0.7 0.6 0.5 Complete PSL setFree PSL 0.4 Attribute Noise 0 0.15 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 1 Structural 0.9 Noise: 0.4 0.8 F1 0.7 0.6 0.5 Complete PSL setFree PSL 0.4 Attribute Noise 0 0.15 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 87
- 88. Probabilistic Soft Logic Decision Making
- 89. Probabilistic Soft Logic Probabilistic Query Analysis Query Type Marginal Distribution Most Probable World continuum Most Probable Sub-World 1 Atom Entire World Examples: Examples: Collective Classification Image Denoising Link Prediction Complex System Configuration Interesting: Constraints Ising Model Decision Level Perspective System Level Perspective 89
- 90. Probabilistic Soft Logic Probabilistic Query Analysis Decision-driven Query Type Marginal Distribution Most Probable World continuum Most Probable Sub-World 1 Atom Entire World Examples: Examples: Collective Classification Image Denoising Link Prediction Complex System Configuration Interesting: Constraints Ising Model Decision Level Perspective System Level Perspective 90
- 91. Probabilistic Soft Logic Probabilistic Query Analysis Decision-driven Query Type Marginal Distribution Most Probable World continuum Most Probable Sub-World 1 Atom Entire World Cannot be Examples: meaningfully Examples: Collective Classification analyzed Image Denoising Link Prediction Complex System Configuration Interesting: Constraints Ising Model Decision Level Perspective System Level Perspective 91
- 92. Probabilistic Soft Logic Probabilistic Query Analysis Query Type Marginal Distribution Most Probable World continuum Most Probable Sub-World 1 Atom Entire World Examples: Decision-driven Collective Classification Examples: Image Denoising Link Prediction Modeling Complex System Configuration Interesting: Constraints Ising Model Decision Level Perspective System Level Perspective 92
- 93. Probabilistic Soft Logic Decision Driven Modeling (DDM) [ 2 ] Predicates are typed as probability distributions - e.g. Bernoulli distributions, parameterized by p ε [0,1] Atoms are RVs over parameterized distributions Defines a second-order probability distribution defined by a CCMRF Allows integration of external classifiers - Important, e.g. in personalized medicine Aggregation of evidence - Can handle sets and other continuous aggregations 93
- 94. Probabilistic Soft Logic Experiments: Wikipedia [3 ] Wikipedia Category Prediction - 2460 featured documents - Links, talks - Predict: category(D,C): Bernoulli - 2 setups: seed split link talk talk link talk talk link talk 94
- 95. Probabilistic Soft Logic Wikipedia Rules hasCat(A,C) ô hasCat(B,C) A!=B unknown(A) document(A,T) document(B,U) similarText(T,U) hasCat(A,C) ô hasCat(B,C) unknown(A) link(A,B) A!=B hasCat(D,C) ô talk(D,A) talk(E,A) hasCat(E,C) unkonwn(D) A!=B 95
- 96. Probabilistic Soft Logic Wikipedia – External Classifier 0.8 0.78 0.76 0.74 0.72 0.7 F1 0.68 0.66 Attributes Only 0.64 Attributes + Links 0.62 Attributes + Links + Talks 0.6 250 375 500 625 750 Number of Training Documents 96
- 97. Probabilistic Soft Logic Wikipedia – Seed Classification 0.7 0.6 0.5 F1 0.4 0.3 0.2 0.15 (220) 0.2 (290) 0.25 (370) 0.3 (440) Percentage of Seed Document (# Documents) Attributes only Attributes + Links Attributes + Links + Talks 97
- 98. Probabilistic Soft Logic Confidence Analysis [5 ] Analyze the confidence in a prediction by computing its marginal density function in the second order probability distribution - What does the density function look like around the MAP state? Novel aspect in SRL f f vs | | | | 0 1 0 1 Category(Doc1,Theory) Category(Doc1,Theory) 98
- 99. Probabilistic Soft Logic Experiments: Confidence Analysis Split predications into S+, S- Compute avg standard deviations for each σ− − σ+ Compare: ∆(σ) = 2 σ+ + σ− Hypothesis: ∆(σ) 0 Folds P(Null Hypothesis) Relative Difference Δ(σ) 20 1.95E-09 38.3% 25 2.40E-13 41.2% 30 1.00E-16 43.5% 35 4.54E-08 39.0% 99
- 100. Probabilistic Soft Logic PSL System Overview
- 101. Probabilistic Soft Logic PSL Implementation Implemented in Java / Groovy Declarative model definition and imperative model interaction ~40k LOC but still alpha Performance oriented - Database backend - Memory efficient data structures - High performance solver integration
- 102. Input Model Probabilistic Rules Similarity Input Data A≈B similarID(A.name,B.name) Graph Preprocessing Logic {A.subClass}≈{B.subClass} A≈B System Overview Constraints RDBMS Partial functional: ≈ Similarity Functions similarID(A,B) = new SimFun(){} Groovy PSL Programming Environment Factor Graph Analysis Grounding Evalua:on Tools Framework Op#miza#on Toolbox Reasoner + Learning Similarity Func#ons Inference Result
- 103. Probabilistic Soft Logic Defining the Model
- 104. Probabilistic Soft Logic Interacting with the Model
- 105. Probabilistic Soft Logic Conclusion Simple and expressive formalism to reason about similarity and uncertainty collectively - Sets aggregates, external functions Scalable due to continuous rather than combinatorial formulation Future: Structure learning, extend framework, additional use cases. 105
- 107. ? Probabilistic Soft Logic Questions? Comments?
- 108. Probabilistic Soft Logic References Bibliography
- 109. Probabilistic Soft Logic Presented Work [5] Computing marginal distributions over continuous Markov networks for statistical relational learning, Matthias Broecheler, and Lise Getoor, Advances in Neural Information Processing Systems (NIPS) 2010 [4] A Scalable Framework for Modeling Competitive Diffusion in Social Networks, Matthias Broecheler, Paulo Shakarian, and V.S. Subrahmanian, International Conference on Social Computing (SocialCom) 2010, Symposium Section [3] Probabilistic Similarity Logic, Matthias Broecheler, Lilyana Mihalkova and Lise Getoor, Conference on Uncertainty in Artificial Intelligence 2010 [2] Decision-Driven Models with Probabilistic Soft Logic, Stephen H. Bach, Matthias Broecheler, Stanley Kok, Lise Getoor, NIPS Workshop on Predictive Models in Personalized Medicine 2010 [1] Probabilistic Similarity Logic, Matthias Broecheler, and Lise Getoor, International Workshop on Statistical Relational Learning 2009 This presentation also covers joint work with Paulo Shakarian and Dr. V.S. Subrahmanian. 109
- 110. Probabilistic Soft Logic References Introduction to Statistical Relational Learning, Lise Getoor and Ben Taskar, MIT Press, 2007 Theory of generalized annotated logic programming and its applications, Michael Kifer and V.S. Subrahmanian, Journal of Logic Programming, Volume 12 Issue 4, April 1992 Using Histograms to Better Answer Queries to Probabilistic Logic Programs, Matthias Broecheler, Gerardo I. Simari, and V.S. Subrahmanian, International Conference on Logic Programming 2009 Hit-and-run from a corner, L. Lovasz and S. Vempala, ACM Symposium on Theory of computing, 2004 110