If correlation doesn’t imply causality, then what does?

Presentation on the Michael Nielsen's insightful article of the same title.

http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/

Transcript

1. IF CORRELATION DOES NOT IMPLY CAUSALITY then what does?
2. REFERENCE “If correlation doesn’t imply causation, then what does?” by Michael Nielsen Quantum physics Quantum computing Machine learning Neural network Deep learning http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/
3. CORRELATION DOES NOT IMPLY CAUSALITY
4. Credit: http://www.bloomberg.com/bw/magazine/correlation-or-causation-12012011-gfx.html
5. Credit: http://www.bloomberg.com/bw/magazine/correlation-or-causation-12012011-gfx.html
6. Credit: http://www.bloomberg.com/bw/magazine/correlation-or-causation-12012011-gfx.html
7. SIMPSON’S PARADOX Correlation can be reversed when you group data differently.
8. WAS BERKELEY SEXIST? Applicants Admitted Men 8442 44% Women 4321 35% Data source: admission for the fall of 1973 at UC Berkeley
9. But if you look closely... Department Men Women Applicants Admitted Applicants Admitted A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7% Total 8442 44% 4321 35%
10. In most departments, admission for women are actually higher than men. Department Men Women Applicants Admitted Applicants Admitted A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7% Total 8442 44% 4321 35%
11. WHY?!
12. Department Men Women Applicants Admitted Applicants Admitted A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7% Total 8442 44% 4321 35% Women tended to apply to competitive departments with low rates of admission
13. 80% of statistics are made up
14. 80% of statistics are made up interpretation ˄
15. I often wonder how many people with real decision-making power – politicians, judges, and so on – are making decisions based on statistical studies, and yet they don’t understand even basic things like Simpson’s paradox. - Michael Nielsen “ ”
16. SO IF CORRELATION DOES NOT IMPLY CAUSALITY then what does?
18. Judea Pearl Causality theory Bayesian networks 2011 Turing Award winner
19. Lung cancer Smoker 17.2% Non-smoker 1.3% AN EXAMPLE
20. smoking cancer causes?
21. TOBACCO COMPANY WOULD SAY smoking cancer hidden genetic factor
22. smoking cancer hidden genetic factor LET’S MAKE IT MORE GENERAL
23. Theoretically, to prove if smoking causes lung cancer, we need to do a randomized controlled experiment.
24. smoking cancer hidden genetic factor RANDOMIZED CONTROLLED EXPERIMENT
25. smoking cancer hidden genetic factor Go smoke (50%) Don’t smoke (50%) RANDOMIZED CONTROLLED EXPERIMENT
26. Lung cancer Smoker P 1 Non-smoker P 2 If P 1 > P 2 , then we can say smoking causes lung cancer from the stats. After the experiment, we measure the percentages of people get cancer:
27. By forcing people (not) to smoke at random, you eliminate the hidden factor that may cause people to smoke. smoking cancer hidden genetic factor Experimenter
28. BUT FORCING PEOPLE IS INHUMAN! BUT FORCING PEOPLE IS INHUMAN!
29. Luckily, Pearl developed a causal calculus allowing us to make inferences without doing any inhuman experiments.
30. Causal models Causal conditional probabilities d-separation 3 rules of causal calculus
31. CAUSAL MODEL (aka Bayesian Network) smoking cancer hidden genetic factor
32. CAUSAL MODEL (aka Bayesian Network) S C H
33. CAUSAL MODEL (aka Bayesian Network) S C H Random variables
34. CAUSAL MODEL (aka Bayesian Network) S C H Influences
35. PARENT NOTATION S C H pa(H) = { } pa(C) = { H, S } pa(S) = { H }
36. PERTURBED GRAPH S C H S C H G G S Remember the “inhuman experiment”?
37. PERTURBED GRAPH S C H S C H G G S
38. Causal models Causal conditional probabilities d-separation 3 rules of causal calculus
39. PROBABILITY REVIEW All people Smokers Hidden gene on Cancer
40. p(s) = S / A All people Smokers Hidden gene on Cancer
41. p(~s) = (A - S) / A All people Smokers Hidden gene on Cancer
42. p(s, c) = (S∩C) / A All people Smokers Hidden gene on Cancer
43. p(c|s) = (S∩C) / S All people Hidden gene on Cancer Smokers
44. DEFINE “CAUSE” do(S) C H p(c|do(s)) > p(c|do(~s)) ⇔ smoking causes cancer
45. Causal models Causal conditional probabilities d-separation 3 rules of causal calculus
46. d-separation If X can tell us about Y, then X and Y are d-connected. Otherwise, they’re d-separated.
47. d-connected X Y
48. d-connected X Y
49. d-separated X ⊥ Y X Y
50. X Y Z (unknown) d-separated X ⊥ Y
51. X Y Z (known) d-connected
52. Causal models Causal conditional probabilities d-separation 3 rules of causal calculus
53. (X ⊥ Y | W, Z) G X Let’s try this notation first. What does it mean?
54. (X ⊥ Y | W, Z) G X Remove all the edges pointing to X in graph G
55. (X ⊥ Y | W, Z) G X Remove all the edges pointing to X in graph G Given that W and Z are known
56. (X ⊥ Y | W, Z) G X Remove all the edges pointing to X in graph G Given that W and Z are known X and Y are d-separated
57. RULE 1 (Y ⊥ Z | W, X) G X p(y|w, do(x), z) = p(y|w, do(x)) ⇒ When can we ignore observations?
58. RULE 1 (Y ⊥ Z | W, X) G X p(y|w, do(x), z) = p(y|w, do(x)) ⇒ When can we ignore observations? Z doesn’t have influence on Y, so we can omit z when calculating p(Y).
59. RULE 2 (Y ⊥ Z | W, X) G X, Z p(y|w, do(x), do(z)) = p(y|w, do(x), z) ⇒ When can we ignore the act of intervention?
60. RULE 2 (Y ⊥ Z | W, X) G X, Z p(y|w, do(x), do(z)) = p(y|w, do(x), z) ⇒ When can we ignore the act of intervention? If removing the influence of Z makes Y and Z unrelated, it makes no difference whether we have intervention on Z or not. Z HAS NO POWER HERE
61. RULE 3 (Y ⊥ Z | W, X) G X, Z - pa(W) p(y|w, do(x), do(z)) = p(y|w, do(x)) ⇒ When can we ignore an intervention variable entirely?
62. RULE 3 (Y ⊥ Z | W, X) G X, Z - pa(W) p(y|w, do(x), do(z)) = p(y|w, do(x)) ⇒ When can we ignore an intervention variable entirely? If Z only affects known variables (X and W), Y has nothing to do with Z.
63. Back to the subject... p(c|do(s)) > p(c|do(~s)) ⇔ smoking causes cancer
64. The 3 rules of causal calculus make it possible to derive p(c|do(s)) and p(c|do(~s)) only from observed statistics. No intervention ➭ no dos!
65. Smoking Cancer Hidden gene Tars in lungs Goal: compute p(c|do(s)) without dos
67. p(c|do(s)) = ? S C H T
68. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) S C H T
69. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) S C H T (T ⊥ S) GS RULE 2
70. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) = Σ t p(c|do(s), do(t))⋅p(t|s) S C H T (C ⊥ T|S) GS, T RULE 2
71. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) = Σ t p(c|do(s), do(t))⋅p(t|s) = Σ t p(c|do(t))⋅p(t|s) S C H T (C ⊥ S|T) GS, T RULE 3
72. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) = Σ t p(c|do(s), do(t))⋅p(t|s) = Σ t p(c|do(t))⋅p(t|s) = Σ t (Σ s p(c|do(t), s)⋅p(s|do(t)))⋅p(t|s) S C H T
73. p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) = Σ t p(c|do(s), do(t))⋅p(t|s) = Σ t p(c|do(t))⋅p(t|s) = Σ t (Σ s p(c|do(t), s)⋅p(s|do(t)))⋅p(t|s) = Σ t (Σ s p(c|t, s)⋅p(s|do(t)))⋅p(t|s) S C H T (C ⊥ T|S) GS, T RULE 2
74. S C H T (S ⊥ T) GT RULE 3 p(c|do(s)) = Σ t p(c|do(s), t)⋅p(t|do(s)) = Σ t p(c|do(s), t)⋅p(t|s) = Σ t p(c|do(s), do(t))⋅p(t|s) = Σ t p(c|do(t))⋅p(t|s) = Σ t (Σ s p(c|do(t), s)⋅p(s|do(t)))⋅p(t|s) = Σ t (Σ s p(c|t, s)⋅p(s|do(t)))⋅p(t|s) = Σ t (Σ s p(c|t, s)⋅p(s))⋅p(t|s)
75. TOY MODEL Tar No tar Smoker 47.5% 85% cancer 2.5% 90% cancer Non-smoker 2.5% 5% cancer 47.5% 10% cancer p(c|do(s)) = 45.25% p(c|do(~s)) = 47.5% ➭ ➭Smoking reduces chance of getting cancer! * * This is not true because the numbers we’re using are not real
76. CONCLUSION
77. Correlation has nothing to do with causality. Remember Simpson’s Paradox.
78. Pearl’s theory of causality can be used to infer causal relationships without experimental intervention. But NOT always!
79. QUESTIONS?