Can one extract causal information from high-dimensional observational data? Applied Multivariate Statistics – Spring 2012 (not relevant for exam)
What is a causal effect? Markus Kalisch, ETH Zurich 2
What is a causal effect? Drowning accidents Markus Kalisch, ETH Zurich 3
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 4
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 5
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 6
What is a causal effect? Drowning accidents Ice cream sales ? Markus Kalisch, ETH Zurich 7
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 8
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 9
What is a causal effect? Drowning accidents Ice cream sales Markus Kalisch, ETH Zurich 10
Another example: Smoking Markus Kalisch, ETH Zurich 11
Scenario 1: Observe 1000 smoker and count the incidence of lung cancer Markus Kalisch, ETH Zurich 12
Scenario 1: Observe 1000 smokers and count the incidence of lung cancer Scenario 2: Make 1000 random people smoke and count the incidence of lung cancer Markus Kalisch, ETH Zurich 13
Scenario 1: Observe 1000 smokers and count the incidence of lung cancer Scenario 2: Make 1000 random people smoke and count the incidence of lung cancer are different. Markus Kalisch, ETH Zurich 14
What is a causal effect? CHANGE BY INTERVENTION Markus Kalisch, ETH Zurich 15
How to find causal effects? Markus Kalisch, ETH Zurich 16
Experimental How to find causal effects? Data ? Markus Kalisch, ETH Zurich 17
Experimental How to find causal effects? Data Two groups of plots: Identical in all aspects (sunlight, water, soil quality, …) Markus Kalisch, ETH Zurich 18
Experimental How to find causal effects? Data Two groups of plots: Identical in all aspects (sunlight, water, soil quality, …) Practice: Randomized assignment Markus Kalisch, ETH Zurich 19
Experimental How to find causal effects? Data Markus Kalisch, ETH Zurich 20
Experimental How to find causal effects? Data Markus Kalisch, ETH Zurich 21
Experimental How to find causal effects? Data Outcome due to fertilizer, since everything else was equal Markus Kalisch, ETH Zurich 22
How to find causal effects? Sometimes, randomized controlled experiments are too expensive (gene experiments) too time-consuming (gene experiments) unethical (HIV treatment) just not practical (smoking). Markus Kalisch, ETH Zurich 23
Observational If experiment is impossible… Data Markus Kalisch, ETH Zurich 24
Observational … observe fields of two farmers. Data Markus Kalisch, ETH Zurich 25
Observational … observe fields of two farmers. Data Groups not guaranteed to be identical in all aspects (sunlight, water, soil quality, …) Markus Kalisch, ETH Zurich 26
Observational … observe fields of two farmers. Data Markus Kalisch, ETH Zurich 27
Observational … observe fields of two farmers. Data Is outcome due to fertilizer? We can’t tell ! Markus Kalisch, ETH Zurich 28
Observational … observe fields of two farmers. Data Markus Kalisch, ETH Zurich 29
Observational … observe fields of two farmers. Data Markus Kalisch, ETH Zurich 30
How to find causal effects? Can one extract causal information from observational data alone? Markus Kalisch, ETH Zurich 31
IDA Goal of this talk IDA finds a set of possible causal effects given observational data consistently even in high dimensions. One element of the set is the true causal effect; bounds on set are useful Does not replace randomized experiments Helps prioritizing and designing random experiments Markus Kalisch, ETH Zurich 32
Example Yeast: Saccharomyces cerevisiae Markus Kalisch, ETH Zurich 33
Example Yeast: Saccharomyces cerevisiae Markus Kalisch, ETH Zurich 34
Example Yeast: Saccharomyces cerevisiae What are the causal effects among the thousands of genes? Markus Kalisch, ETH Zurich 35
Example Yeast: Saccharomyces cerevisiae What are the causal effects among the thousands of genes? Approach: Model gene expression of each gene as a random variable. Can we use the joint distribution of gene expression to extract causal information? Markus Kalisch, ETH Zurich 36
Here is a distribution oracle. Now find the causal effect! Distribution oracle Markus Kalisch, ETH Zurich 37
Outline in Theory Distribution oracle do-calculus Causal IDA with known Structure causal structure Causal effects Markus Kalisch, ETH Zurich 38
do-calculus Pearl’s do -operator with known causal structure Notation for causal intervention P(Y=y | do(X=x)) “distribution of Y, if there is an intervention in variable X ” Causal effect C(x’) = d/ dx E[Y=y | do(X=x)]| x=x’ “change in expected value of Y, if there is an intervention in variable X ” Markus Kalisch, ETH Zurich 39
do-calculus P(Y=y | X=x) ≠ P(Y=y | do(X=x)) with known causal structure Pick a random day: P(rain | wet) = high P(rain | do(wet)) = = P(rain) = = low Markus Kalisch, ETH Zurich 40
do-calculus Pearl’s do -calculus with known causal structure Causal structure Rules: Z Expression with “do” X Expression without “do” Y Judea Pearl, “ Causality” , 2010, Cambridge University Press Markus Kalisch, ETH Zurich 41
do-calculus with known Example: Back-door Adjustment causal structure Assume Z is binary (0/1) Causal structure P(Y=y | do(X=x)) Z Rules X P(Y=y | X=x, Z=0) * P(Z=0) + P(Y=y | X=x, Z=1) * P(Z=1) Y Markus Kalisch, ETH Zurich 42
do-calculus with known Example: Back-door Adjustment causal structure Assume Z is binary (0/1) “do” Causal structure P(Y=y | do(X=x)) Z Rules X P(Y=y | X=x, Z=0) * P(Z=0) + P(Y=y | X=x, Z=1) * P(Z=1) Y Markus Kalisch, ETH Zurich 43
do-calculus with known Example: Back-door Adjustment causal structure Assume Z is binary (0/1) Causal structure P(Y=y | do(X=x)) Z Rules X P(Y=y | X=x, Z=0) * P(Z=0) + P(Y=y | X=x, Z=1) * P(Z=1) Y No “do” Markus Kalisch, ETH Zurich 44
do-calculus with known Conclusion 1 causal structure If causal structure is known, we can infer causal effects from observations Markus Kalisch, ETH Zurich 45
Outline in Theory Distribution IDA oracle do-calculus Causal with known Structure causal structure Causal effects Markus Kalisch, ETH Zurich 46
Causal Structure Estimate Causal Structure Oftentimes, causal structure is unknown Estimate causal structure Markus Kalisch, ETH Zurich 47
Causal Structure Causal Directed Acyclic Graph (DAG) X W Z Y Markus Kalisch, ETH Zurich 48
Causal Structure Causal Directed Acyclic Graph (DAG) X W Random Direct Variables cause Z Y Markus Kalisch, ETH Zurich 49
Causal Structure Causal Directed Acyclic Graph (DAG) X W Random Direct Variables cause Z Y implies Conditional independence relations among variables Markus Kalisch, ETH Zurich 50
Causal Structure Estimate a DAG model DAG encodes independence information Independencies among Reverse DAG variables given engineering by oracle Markus Kalisch, ETH Zurich 51
Causal Structure Estimate a DAG model DAG encodes independence information PC Algorithm Independencies among Reverse DAG variables given engineering by oracle P. Spirtes, C. Glymour, R. Scheines , “ Causation, Prediction, and Search ”, 2000, MIT Press Markus Kalisch, ETH Zurich 52
Causal Structure Ambiguity: Equivalence class Several DAGs describe exactly the same list of independence relations X W X W Y Z Y Z Markus Kalisch, ETH Zurich 53
Causal Structure Ambiguity: Equivalence class Several DAGs describe exactly the same list of independence relations X W X W Y Z Y Z Markus Kalisch, ETH Zurich 54
Causal Structure Ambiguity: Equivalence class Several DAGs describe exactly the same list of independence relations X W X W Y Z Y Z X W Y Z Equivalence class: PARTIALLY Directed Acyclic Graph (PDAG) Markus Kalisch, ETH Zurich 55
Causal Structure Ambiguity: Equivalence class Several DAGs describe exactly the same list of independence relations X W X W Y Z Y Z X W Y Z Equivalence class: PARTIALLY Directed Acyclic Graph (PDAG) Markus Kalisch, ETH Zurich 56
Causal Structure Ambiguity: Equivalence class Some DAGs describe exactly the same list of independence relations X W X W Y Z Y Z X W PC Algorithm finds equivalence class Y Z Equivalence class: PARTIALLY Directed Acyclic Graph (PDAG) Markus Kalisch, ETH Zurich 57
Outline in Theory Distribution IDA oracle do-calculus Causal with known Structure causal structure Up to equivalence class Causal effects Markus Kalisch, ETH Zurich 58
Putting everything together DAG 1 Effect 1 Distribution PDAG … Set of causal effects oracle DAG n Effect n Markus Kalisch, ETH Zurich 59
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