Intro to Causal Modeling

Transcript

1. 1 CAUSAL MODELING ROBERT SUHADA suhada.robert@gmail.com Data Geeks Meetup
2. 2 TABLE OF CONTENTS Motivation Correlation is not causation Paradox of statistical inference The machinery of causal calculus Probabilistic graphical models Causal conditional probabilities d-separation The 3 Rules of Causal Calculus Does smoking cause cancer? Predict or Explain? Summary Questions? Appendix
3. 3 . 1 MOTIVATION
4. 4 . 1 CORRELATION IS NOT CAUSATION
5. 5 . 1 PIRATES!
6. 6 . 1 NICOLAS CAGE!
7. 7 . 1 MURDER!
8. 8 . 1 DOES CORRELATION IMPLY CAUSATION?
9. 9 . 1 DOES CORRELATION IMPLY CAUSATION? A causes B B causes A Correlation by random chance or selection bias A and B are caused by C
10. 10 . 1 PARADOX OF STATISTICAL INFERENCE
11. 11 . 1 THE PARADOX Statistics can tell you , p(disease|symptom) but never that symptoms do not cause disease.
12. 12 . 1 THE PARADOX There are thousands of people paid to nd causal relationships and all they can deliver are correlations! – Judea Pearl (paraphrased)
13. 13 . 1 DOES IT EVEN MATTER? Better overall: Treatment B
14. 14 . 1 IT MATTERS A LOT! Better overall: Treatment B Better for small stones: Treatment A Better for large stones: Treatment A
15. 15 . 1 CONTINUOUS CASE OF SIMPSON'S PARADOX
16. 16 . 1 INTERVENING VARIABLE LEADS TO REVERSAL
17. 17 . 1 RANDOMIZED CONTROLLED TRIALS a special case is A/B testing
18. 18 . 1 THE PROBLEM WITH RCTS 1. Can address only a subset of interesting questions 2. Costly, take a lot of time 3. Can be trickier proceduraly than one thinks 4. Sometimes impossible practically or ethically…
19. 19 . 1 PROMISE OF CAUSAL MODELLING Given a data set, it is possible to constrain certain causal relationships. We can tell the effect of real-world interventions without actually doing them.
20. 20 . 1 THE MACHINERY OF CAUSAL CALCULUS 1. De ne 3 tools of causal calculus 2. Derive 3 rules of causal calculus
21. 21 . 1 THE 3 TOOLS 1. Probabilistic graphical models 2. Causal conditional probabilities ("do" notation) 3. d-separation
22. 22 . 1 PROBABILISTIC GRAPHICAL MODELS
23. 23 . 1 JOINT PROBABILITY DISTRIBUTION As a good Bayesian, I'll just write down the full joint probability distribution of the model. p( , …, , …, ) x 1 x j x n
24. 24 . 1 JOINT PROBABILITY DISTRIBUTION
25. 25 . 1 JOINT PROBABILITY DISTRIBUTION Smoking Tar Cancer % T T T .000162 T T F .0000085 T F T .0151 T F F .00168 F T T .0078 F T F .002 F F T .00097 F F F .972
26. 26 . 1 PROBLEM: SCALING Joint probability functions scope scales at minimum as ! 2 n Q: How can we ef ciently and sparsely encode all this information? A: Graphical models!
27. 27 . 1 GRAPHICAL MODELS Value of given node = ( ) X j f j Xpa(j) The graph's joint probability function p( , , …) = p( |pa( )) x 1 x 2 ∏ j x j x j
28. 28 . 1 CAUSAL CONDITIONAL PROBABILITIES
29. 29 . 1 THE "DO" NOTATION We don't / can't do a RCT, so let's "simulate" an experimental intervention: p( , …, , …, |do( )) = x 1 x ̂ j x n x j p( , …, ) x1 xn p( |pa( )) xj xj
30. 30 . 1 DEFINITION OF CAUSAL INFLUENCE We say X has a causal in uence over Y, iff there are values and of X and y of Y such that x 1 x 2 p(y|do( )) ≠ p(y|do( )) x 1 x 2
31. 31 . 1 D-SEPARATION
32. 32 . 1 INFORMAL DEFINITION X and Y are d-separated on a graph G if a X can’t tell us anything about the value of another random variable Y in the model, or vice versa. Otherwise they are d-connected.
33. 33 . 1 TRAVERSE d-connected
34. 34 . 1 CONDITIONAL TRAVERSE d-separated parents screen-off the in uence of the more remote ancestors
35. 35 . 1 FORK d-connected
36. 36 . 1 CONDITIONAL FORK d-separated
37. 37 . 1 COLLIDER d-separated
38. 38 . 1 CONDITIONAL COLLIDER d-connected
39. 39 . 1 THE 3 RULES OF CAUSAL CALCULUS
40. 40 . 1 RULE 1: IGNORING OBSERVATIONS Suppose . (Y ⊥ Z|W, X) G X ⎯⎯ ⎯ Then: . p(y|w, do(x), z) = p(y|w, do(x))
41. 41 . 1 RULE 2: IGNORING THE ACT OF INTERVENTION Suppose . (Y ⊥ Z|W, X) G , X ⎯⎯ ⎯ Z ⎯⎯ ⎯ Then: . p(y|w, do(x), do(z)) = p(y|w, do(x), z)
42. 42 . 1 RULE 3: IGNORING AN INTERVENTION VARIABLE ENTIRELY Let denote the set of nodes in Z which are not ancestors of W. Z(W) Suppose . (Y ⊥ Z|W, X) G , X ⎯⎯ ⎯ Z(W) ⎯ ⎯ ⎯⎯⎯⎯⎯⎯⎯⎯ Then: . p(y|w, do(x), do(z)) = p(y|w, do(x))
43. 43 . 1 PROOF The proof is left as an exercise to the reader :)
44. 44 . 1 DOES SMOKING CAUSE CANCER?
45. 45 . 1 CORRELATION CLAIM
46. 46 . 1 CAUSATION CLAIM
47. 47 . 1 STATISTICS GOES TO CONGRESS
48. 48 . 1 MODEL 1: ANTI-SMOKING PROPAGANDA
49. 49 . 1 MODEL 2: GENETICS?
50. 50 . 1 MODEL 3: TOBACCO COMPANY FAVOURITE
51. 51 . 1 SMOKING CAUSES CANCER? Armed with the 3 tools and the 3 rules, we can now: 1. Collect Smoker / non-smoker observational data Look Ma' no RCT data! 2. Draw the graphical causal model 3. Derive p(cancer|do(smoke))
52. 52 . 1 THE DATA 1 Note: all this purely type of data Tobacco companies: smoking in both tar & non-tar groups -> lower cancer rate p(X|Y) Summary: Smoking is GOOD for you!
53. 53 . 1 THE DATA 2 Note: all this purely type of data Anti-smoking lobby: 1. tar increases cancer in both groups (+5 p. points) 2. smoking increases tar (380/400 vs. 20/380) p(X|Y) Summary: Smoking is BAD for you!
54. 54 . 1 THE CAUSAL MODEL
55. 55 . 1 RESULT < < I n s e r t f u r i o u s h a n d - w a v i n g > > , where p(y|do(x)) = p(y| , z)p(z|x)p( ) ∑ z x ′ x′ x′ p(y|do(x)) = p(cancer|do(smoke)) The result is: smoking causes tar deposits and those increase cancer! (note: data is made up) Summary: Smoking is BAD for you!
56. 56 . 1 WHY NOT THE SIMPLER MODEL?
57. 57 . 1 WHERE DID THE GRAPH COME FROM Makes assumption explicit Domain expert illication Graph structure learning No causes in, no causes out! – Nancy Cartwright
58. 58 . 1 PREDICT OR EXPLAIN?
59. 59 . 1 CORRELATION IS OFTEN ENOUGH FOR PREDICTION Image Source: David Manheim
60. 60 . 1 DNN AND CAUSALITY Link: Yannir Serousi
61. 61 . 1 DNN AND CAUSALITY Object-Context Segmentation & observational causal signals See also Athley's work with Random Forests. Discovering Causal Signals in Images, arxiv.org/pdf/1605.08179v1.pdf
62. 62 . 1 SUMMARY
63. 63 . 1 TAKE-AWAY: OBJECT-LEVEL It is possible to constrain certain causal relationships from observational data. The tool for that is Causal Calculus.
64. 64 . 1 TAKE-AWAY: META-LEVEL I beseech you, in the bowels of Christ, think it possible that you may be mistaken. – Oliver Cromvell
65. 65 . 1 QUESTIONS? Judea Pearl Further reading: Michael Nielsen: Eliezer Yudkowsky: If correlation doesn’t imply causation, then what does? Causal Diagrams and Causal Models
66. 66 . 1 APPENDIX
67. 67 . 1 ALTERNATIVES There are many ways about how to solve the prob. graphical models There are also completely different approaches Neyman-Rubin Causality Granger Causation coef cient The "econometrics toolkit": Structural Models / instrumental variables etc.
68. 68 . 1 LINKS: FUN http://guessthecorrelation.com/index.html http://www.tylervigen.com/spurious-correlations https://www.google.com/trends/explore#q=global%20warming,%20pirates&d m&cmpt=q&tz=Etc/GMT-2