JOINT PROBABILISTIC INFERENCE OF CAUSAL STRUCTURE

Transcription

1 JOINT PROBABILISTIC INFERENCE OF CAUSAL STRUCTURE Dhanya Sridhar Lise Getoor U.C. Santa Cruz KDD Workshop on Causal Discovery August 14 th,

2 Outline Motivation Problem Formulation Our Approach Preliminary Results 2

3 Traditional to Hybrid Approaches X 1 X 2 X 1 X 2 Score(G, D) X X X 3 X X 4 Constraint Based X 1 X 2 Score(G, D) X 3 Search and Score Based 3

4 Traditional to Hybrid Approaches X 1 X 2 X 1 X 2 Score(G, D) X X X 3 X X 4 Constraint Based X 1 X 2 Score(G, D) X 4 Search and Score Based Hybrid Approaches 4

5 Traditional to Hybrid Approaches X 1 X 2 X 1 X 2 X X X 3 X X 4 X 3 X 4 Constraint Based Score(G, D) Hybrid Approaches: PC-based DAG Search Dash and Drudzel, UAI 99 Min-max Hill Climbing Tsamardinos et al., JMLR 06 5

6 Joint Inference for Structure Discovery Joint Inference of Variables: X 1 C 12 X 2 Causal Edge C ij Adjacency Edges A ij A 13 A 34 C 24 X 3 X 4

7 Joint Inference for Structure Discovery Joint Inference of Variables: X 1 C 12 X 2 Causal Edge C ij Adjacency Edges A ij A 13 A 34 C 24 X 3 X 4 Joint Inference Approaches: Linear Programming Relaxations, Jaakkola et al., AISTATS 10

8 Joint Inference for Structure Discovery Joint Inference of Variables: X 1 C 12 X 2 Causal Edge C ij Adjacency Edges A ij A 13 A 34 C 24 X 3 X 4 Joint Inference Approaches: Linear Programming Relaxations, Jaakkola et al., AISTATS 10 MAX-SAT, Hyttinen et al., UAI 13

9 Outline Motivation Problem Formulation Our Approach Preliminary Results 9

10 Probabilistic Joint Model of Causal Structure C 12 X 1 X 2 A 13 A 34 C 24 X 3 X 4 Extending joint approaches: probabilistic model over causal structures

11 Probabilistic Joint Model of Causal Structure C 12 X 1 X 2 A 13 A 34 C 24 X 3 X 4 Independence Tests

12 Probabilistic Joint Model of Causal Structure C 12 X 1 X 2 A 13 A 34 C 24 X 3 X 4 Combining logical and structural constraints and probabilistic reasoning

13 Outline Motivation Problem Formulation Our Approach Preliminary Results 13

14 Probabilistic Soft Logic (PSL) Logic-like syntax with probabilistic, soft constraints Describes an undirected graphical model 5.0: Causes(A, B) ^ Causes(B, C) ^ Linked(A,C) à Causes(A, C) Weighted rules Bach et. al (2015). Hinge-loss Markov Random Fields and Probabilistic Soft Logic. arxiv. Open source software: 14

15 Probabilistic Soft Logic (PSL) Logic-like syntax with probabilistic, soft constraints Describes an undirected graphical model 5.0: Causes(A, B) ^ Causes(B, C) ^ Linked(A,C) à Causes(A, C) Weighted rules Predicates are continuous random variables! Bach et. al (2015), arxiv Open source software: 15

16 Probabilistic Soft Logic (PSL) Logic-like syntax with probabilistic, soft constraints Describes an undirected graphical model Relaxations of Logical Operators 5.0: Causes(A, B) ^ Causes(B, C) ^ Linked(A,C) à Causes(A, C) Weighted rules Predicates are continuous random variables! Bach et. al (2015), arxiv Open source software: 16

17 Probabilistic Soft Logic (PSL) Rules instantiated with values from real network 5.0: Causes(A, B) ^ Causes(B, C) ^ Linked(A,C) à Causes(A, C) C 12 X 1 X 2 A 13 A 34 C 24 X 3 X 4 17

18 Probabilistic Soft Logic (PSL) Rules instantiated with variables from real network C : Causes(X 1, X 2 ) ^ Causes(X 2, X 4 ) ^ Linked(X 1,X 4 ) à Causes(X 1, X 4 ) C 12 C 14 A 14 18

19 Soft Logic Relaxation 5.0: Causes(X 1, X 2 ) ^ Causes(X 2, X 4 ) ^ Linked(X 1,X 4 ) à Causes(X 1, X 4 ) Convex relaxation of implication and distance to rule satisfaction Linear Function Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 19

20 Hinge-loss Markov Random Fields p(y X) = 2 1 Z(w, X) exp 4 mx j=1 3 h w j max { j (Y, X), 0}] {1,2}i 5 Conditional random field Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 20

21 Hinge-loss Markov random fields p(y X) = 2 1 Z(w, X) exp 4 mx j=1 3 h w j max { j (Y, X), 0}] {1,2}i 5 Conditional random field Feature functions are hinge-loss functions Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 21

22 Hinge-loss Markov random fields p(y X) = 2 1 Z(w, X) exp 4 mx j=1 3 h w j max { j (Y, X), 0}] {1,2}i 5 Conditional random field Feature function for each instantiated rule Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 22

23 Hinge-loss Markov random fields p(y X) = 2 1 Z(w, X) exp 4 mx j=1 3 h w j max { j (Y, X), 0}] {1,2}i 5 Conditional random field 5.0: Causes(X 1, X 2 ) ^ Causes(X 2, X 4 ) ^ Linked(X 1,X 4 ) à Causes(X 1, X 4 ) Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 23

24 Hinge-loss Markov random fields p(y X) = 2 1 Z(w, X) exp 4 mx j=1 3 h w j max { j (Y, X), 0}] {1,2}i 5 Conditional random field MAP Inference Intuition: minimize distances to satisfaction! Bach et al. (2015), arxiv Bach et al. NIPS 12, Bach et al. UAI 13 24

25 Fast Inference in Hinge-loss MRFs Convex, continuous inference objective Convex optimization! Solved using efficient, message-passing algorithm called Alternating Direction Method of Multipliers Algorithms for weight learning and reasoning with latent variables Bach et al. (2015), arxiv Open source software: Bach et al. NIPS 12, Bach et al. UAI 13 25

26 Encoding PC Algorithm with PSL PC Algorithm: No latent variables and confounders Constraint-based approach PC with PSL: Use all independence tests All rule weights set to

27 PSL Causal Structure Discovery Multiple independence tests with various separation sets No early pruning! 27

28 PSL Causal Structure Discovery Colliders in triples using d-separation 28

29 PSL Causal Structure Discovery 29

30 PSL Causal Structure Discovery 30

31 PSL Causal Structure Discovery 31

32 Outline Motivation Problem Formulation Our Approach Preliminary Results 32

33 Evaluation Dataset Synthetic Causal DAG Dataset 2000 examples Causality Challenge: 33

34 Evaluation Experimental setup: G 2 Independence Tests for both PC and PSL Max separation set of size 3 Evaluation details Run PC and PC-PSL algorithms and compare to causal ground truth For PSL, round with threshold selected by crossvalidation on causal edges 34

35 Causal Edge Prediction Results Average causal edge prediction accuracy and F1 score on 3-fold cross validation Accuracy F1 Score PC Algorithm 0.91 ± ± 0.26 PC-PSL 0.94 ± ±

36 Summary and Future Directions Joint inference of causal structure using probabilistic, soft constraints Incorporate prior and domain knowledge for causal edges from text-mining, ontological constraints, and variable selection methods Extensive, cross-validation experiments on multiple datasets 36