Estimate intervention effects – Need causal graph to select variables to be adjusted, e.g., using backdoor criterion (Pearl, 1995) • Also useful for machine learning – E.g., domain adaptation (Zhang et al., 2020), fairness (Kuzner et al., 2017), and interpretability (Blobaum & Shimizu, 2017) 3 Messerli (2012) Chocolate Nobel laureates GDP Number of Nobel laureates Chocolate consumption
Use background knowledge • Often need to use both background knowledge AND DATA • Causal discovery: Infer the causal graph from data 4 ? or or Chocolate Nobel laureates GDP Chocolate Nobel GDP Chocolate Nobel GDP Chocolate Nobel GDP
non-parametric approach uses conditional independence (Pearl 2001; Spirtes 1993) – Make no assumptions about function forms or distribution – The limit is finding the Markov equivalent models • Additional assumptions needed to go beyond the limit – Restrictions on functional forms and distributions – Uniquely Identifiable or Smaller numbers of Equivalent models • LiNGAM is one example (Shimizu et al., 2006; Shimizu, 2014). – Non-Gaussian assumption to exploit independence – Growing literature on its variants (Peters et al., 2018; Shimizu & Blobaum, 2020) 6
non-parametric approach uses conditional independence (Pearl 2001; Spirtes 1993) – Make no assumptions about function forms or distribution – The limit is finding the Markov equivalent models • Additional assumptions needed to go beyond the limit – Restrictions on functional forms and distributions – Uniquely identifiable or smaller numbers of equivalent models • LiNGAM is one example (Shimizu et al., 2006; Shimizu, 2014). – Non-Gaussian assumption to exploit independence – Growing literature on its variants (Peters et al., 2018; Shimizu & Blobaum, 2020) 7
non-parametric approach uses conditional independence (Pearl 2001; Spirtes 1993) – Make no assumptions about function forms or distribution – The limit is finding the Markov equivalent models • Additional assumptions needed to go beyond the limit – Restrictions on functional forms and distributions – Uniquely identifiable or smaller numbers of equivalent models • LiNGAM is one example (Shimizu et al., 2006; Shimizu, 2014). – Non-Gaussian assumption to exploit independence – Growing literature on its variants (Peters et al., 2018; Shimizu & Blobaum, 2020) 8
and find a causal graph(s) that is consistent with the data – Typical example 1: • Directed acyclic graph (DAG) • No hidden common cause (all observed) – Typical example 2: • DAG • Hidden common causes may exist 10 x3 x1 e3 e1 x2 e2 Error variable 𝑥! = 𝑓! (parents of 𝑥! , 𝑒! )
graph – Directed acyclic graph – No hidden common causes (all have been observed) 2. Find the graph that best matches the data among such causal graphs that satisfy the assumptions. 12 If x and y are independent in the data, select (c) on the right. If x and y are dependent in the data, select (a) and (b). (a) and (b) are indistinguishable (not uniquely identifiable): Markov equivalence class Three candidates x y x y x y (a) (b) (c)
graph – Directed acyclic graph – No hidden common causes (all have been observed) 2. Find the graph that best matches the data among such causal graphs that satisfy the assumptions. 13 If x and y are independent in the data, select (c) on the right. If x and y are dependent in the data, select (a) and (b). (a) and (b) are indistinguishable (not uniquely identifiable): Markov equivalence class Three candidates x y x y x y (a) (b) (c)
et al., 1995) • Those for time series cases (Malinsky & Spirtes, 2018) • Equivalence class including cyclic graphs (Richardson, 1996) • Lower bound on intervention effects (Maathuis et al., 2009; Malinsky & Spirtes, 2017) 14 x y ｆ w z x y w z x y ｆ1 w z ｆ2 F. Eberhardt CRM Workshop 2016
information available than conditional independence • E.g., linearity + non-Gaussian continuous distribution 16 Results in different distributions of x1 and x2 No difference in terms of their conditional independence x y x y (a) (b)
independence of error variables, e.g., by HSIC (Gretton et al., 2005) – Prediction accuracy using Markov boundary (Biza et al., 2020) – Compare to the results of other datasets in which causal graphs expected to be similar – Check against background knowledge 22
Peters+14JMLR) • 𝑥% = 𝑓%(par(𝑥%)) + 𝑒% • 𝑥% = 𝑔% &"(𝑓%(par(𝑥%)) + 𝑒%) • Discrete variables – Poisson DAG model and its extensions (Park+18JMLR) • Mixed types of variables: LiNGAM + logistic-type model – Identifiability condition for two variables (Wenjuan+18IJCAI) – Probably ok also for multivariate cases using the idea of Thm.28 of Peters et al. (2014) 25
Peters+14JMLR) • 𝑥% = 𝑓%(par(𝑥%)) + 𝑒% • 𝑥% = 𝑔% &"(𝑓%(par(𝑥%)) + 𝑒%) • Discrete variables – Poisson DAG model and its extensions (Park+18JMLR) • Mixed types of variables: LiNGAM + logistic-type model – Identifiability condition for two variables (Wenjuan+18IJCAI) – Probably ok also for multivariate cases using the idea of Thm.28 of Peters et al. (2014) 26
common causes • For unconfounded pairs with no hidden common causes, estimate the causal directions • For confounded pairs with hidden common causes, leave them remain unknown 32 𝑥# 𝑥" 𝑓" 𝑥$ Underlying model Output 𝑥0 𝑥# 𝑥" 𝑥$ 𝑥0 𝑓#
causes leads to dependence btw. explanatory variable and its residual (Tashiro et al., 2014) • Key result (Maeda & Shimizu, 2020) – Find a set of variables that that gives independent residual when a variable is regressed on every its subset – If succeeded, variables in such a set (x1 and x2) are the unconfounded ancestors of the variable (x4) • For nonlinear additive models, existence of hidden intermediate variables also leads to dependence (Maeda & Shimizu, 2021) 33 𝑥# 𝑥" 𝑓" !! !" "" !# !$ "! !! 𝑥# 𝑥" 𝑓$
causes leads to dependence btw. explanatory variable and its residual (Tashiro et al., 2014) • Key result (Maeda & Shimizu, 2020) – Find a set of variables that that gives independent residual when a variable is regressed on every its subset – If succeeded, variables in such a set (x1 and x2) are unconfounded ancestors of the variable (x4) • For nonlinear additive models, existence of hidden intermediate variables also leads to dependence (Maeda & Shimizu, 2021) 34 𝑥# 𝑥" 𝑓" !! !" "" !# !$ "! !! 𝑥# 𝑥" 𝑓$
causes leads to dependence btw. explanatory variable and its residual (Tashiro et al., 2014) • Key result (Maeda & Shimizu, 2020) – Find a set of variables that that gives independent residual when a variable is regressed on every its subset – If succeeded, variables in such a set (x1 and x2) are unconfounded ancestors of the variable (x4) • For nonlinear additive models, existence of hidden intermediate variables also leads to dependence (Maeda & Shimizu, 2021) 35 𝑥# 𝑥" 𝑓" !! !" "" !# !$ "! !! 𝑥# 𝑥" 𝑓$
– 2 pure measurement variables per latent needed to identify the measurement model (Silva et al., 2006; Xie et al., 2020) • Estimate the latent factors and then their causal graph 39 𝑥" 𝑥! $ 𝑓" $ 𝑓! 𝑥# 𝑥$ ? 𝒇 = 𝐵𝒇+𝝐 𝒙 = 𝐺𝒇+𝒆
for science – Many well-developed methods available in cases that a causal graph can be drawn with background knowledge – Helping drawing causal graphs with data is the key: Causal discovery • LiNGAM-related papers: https://sites.google.com/view/sshimizu06/lingam/lingampapers • Next default assumptions – Hidden common cause / latent factors – Mixed data: Continuous and discrete – (Cyclicity (Lacerda et al., 2008)) 42
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