Models of Causality. Roy Dong. University of California, Berkeley

Transcription

1 Models of Causality Roy Dong University of California, Berkeley

2 Correlation is not the same as causation. 2

3 Conditioning is not the same as imputing. 3

4 Stylized example Suppose, amongst the population, we have the following simple model: Income: X " User owns eco-friendly fridge: X # Monthly energy consumption: X $ Income Eco-friendly refrigerator? Monthly energy consumption 4

5 Stylized example What would a rebate for free eco-friendly fridges to the population do? Intuitive approach: Take empirical data and learn f: X # X $. Then compare f X # = 1 and f(x # = 0). Note the implicit conditioning! Income Eco-friendly refrigerator? Monthly energy consumption 5

6 Income: Stylized example X " ~Uniform 0,1 User owns eco-friendly fridge: X # X " ~Bernoulli X " Monthly energy consumption: X $ X ", X # ~Normal 1 + 2X " 1 2 X #, σ # Income Eco-friendly refrigerator? Monthly energy consumption 6

7 Stylized example X " ~Uniform 0,1 X # X " ~Bernoulli X " X $ X ", X # ~Normal 1 + 2X " 1 2 X #, σ # Intuitive approach yields: f X # = 1 = 11 6, f X # = 0 = 5 3 Income Effect of X # is " D. Eco-friendly refrigerator? Monthly energy consumption 7

8 Stylized example Causal effects do not go upstream. Giving someone an eco-friendly fridge does not change their income. To look at the causal effects, we impute on the node. Income Eco-friendly refrigerator? Monthly energy consumption 8

9 Stylized example Causal approach yields: Effect of X # is " #. Compare with the intuitive approach: Effect of X # is " D. Income Eco-friendly refrigerator? Monthly energy consumption 9

10 Previous literature Three main paradigms for formalizing causality: Rubin causality Granger causality Pearl s causality 10

11 Rubin causality There is some control variable X 0,1. Example: X = 1 means I am given a new medication. X = 0 means I am given a placebo. There are two distinct random variables Y " and Y I which are the outcome when X = 1 or X = 0. Example: Y " is my blood pressure when I am given my medication. Y I is my blood pressure when I am given a placebo. Fundamental misery of causality: I only observe either Y " or Y I, but not both. [Rubin 1974] [Imbens and Rubin 2015] [Athey and Imbens 2015] [Zhou, Balandat, and Tomlin 2015] 11

12 Rubin causality [Rubin 1974] [Imbens and Rubin 2015] [Athey and Imbens 2015] [Zhou, Balandat, and Tomlin 2015] 12

13 Granger causality Two stationary stochastic processes X and Y. Let: U K denote all the information in the universe available at time t. U X K denote all the information in the universe available at time t except X. σ # (Y U) is the variance of the unbiased, least-squares estimator of Y using U. (Similarly σ # (Y U X).) [Granger 1969] [Brovelli, Ding, Ledberg, Chen, Nakamura, and Bressler 2004] [Gupta and Mazumdar 2014] 13

14 Granger causality U K denote all the information in the universe available at time t. U X K denote all the information in the universe available at time t except X. σ # (Y U) is the variance of the unbiased, least-squares estimator of Y using U. (Similarly σ # (Y U X).) Then X Granger-causes Y if: σ # Y U < σ # Y U X [Granger 1969] [Brovelli, Ding, Ledberg, Chen, Nakamura, and Bressler 2004] [Gupta and Mazumdar 2014] 14

15 Granger causality X Y Time 15

16 Pearl s causality Suppose we have a Bayesian network. We impute a node X N to value x by disconnecting it from all its parents, and setting its value to x. The causal power of X N over X P is determined by how much the distribution of X P changes due to imputation on X N. [Pearl 2000] 16

17 Pearl s causality do(x; i, x) i i X Y [Dong, Mazumdar, Sastry, Optimal Causal Imputation for Control IEEE CDC (in preparation).] 17

18 Previous literature Three main paradigms for formalizing causality: Rubin causality Focused on estimation of the counterfactual. Granger causality Focused on explanatory power. Pearl s causality Focused on imputation. 18

19 Model Given: Directed acyclic graph (DAG) which consists of nodes G and directed edges E G G. Random process X indexed by G which factorizes: P X = Z P X N pa X N N ^ This has many names: X and G are compatible. G represents X. X is Markov relative to G. Interpretation: If there is an edge going from i to j, then we say X N causes [Pearl 2000] X P. 19

20 Learning causal structures Throughout this talk, I will assume this causal structure G, E is given. This is a non-trivial! An active topic of research: Using prior knowledge Econometrics System identification techniques Message passing algorithms Frequency domain analysis of time signals Chao-Liu trees and directed information methods &c 20

21 Definition of imputation By disintegration results in probability theory, if all the random elements X N live in Borel spaces, we can equivalently write: X N = f N pa X N, ξ N where ξ N ~Uniform[0,1] are independent [Pearl 2000] [Kallenberg 2002, Chapter 5] [Dong, Mazumdar, Sastry, Optimal Causal Imputation for Control IEEE CDC (in preparation).] 21

22 Definition of imputation We say a random process Y indexed by G is the imputation of X at i G to a constant x if: At i: Y N = x For all j that is not a descendent of i: Y P = X P For any j that is a descendant of i: Y P = f P pa Y P, ξ P i We write: Y = do X; i, x 22

23 Definition of imputation do(x; i, x) i i X Y 23

24 Definition of imputation We say a random process Y indexed by G is the imputation of X at i G to a constant x if: At i: Y N = x For all j that is not a descendent of i: Y P = X P For any j that is a descendant of i: Y P = f P pa Y P, ξ P i We write: Y = do X; i, x 24

25 Closing remarks In order to close the loop around analytics, we need to move from predictive estimators to causal structures. Estimation of causal structures is half of the problem; control of causal structures is important as well. 25