Shimizu, 2014) • Classic methods use conditional independence of variables (Pearl 2001; Spirtes 1993) – The limit is finding the Markov equivalent models • Need more assumptions to go beyond the limit – Restrictions on the functional forms or/and the distributions of variables • LiNGAM is an example – non-Gaussian assumption to examine independence – Unique identification or smaller numbers of equivalent models 3
causes • ICA: Independent Component Analysis (Comon, 1991; Eriksson et al., 2004; Hyvarinen et al., 2001) – Factor analysis with no factor rotation indeterminacy – Factors are independent and non-Gaussian • LiNGAM with hidden common causes is ICA 8 = + + = ( − )(& ( − )(& LiNGAM with hidden common causes ICA
Washio, 2014) • Do the following for all the variables - ( = 1, … , ) – Regress - on the other variables – If and only if the explanatory variables and residual are independent, the variable is an unconfounded sink • Exclude the sink • Repeat … 12 !! !" "" !# !$ !! !" "" !# !! !" "" The algorithm stops #$ ##
& Shimizu, 2020) • 1. Find unconfounded ancestors of each variable • 2. Find unconfounded parents among the unconfounded ancestors found 13 Find a set of variables that gives independent residuals when # is regressed on every its subset (Lemma 3) Regress # on the unconfounded ancestors of # except ! Regress ! on the unconfounded common ancestors of ! and # If the two residuals are correlated, ! is a (unconfounded) parent of ! Otherwise not (Lemma 4) Wang and Drton (2020, arXiv preprint) considered criteria that can be applied to more general cases !! !" "" !# !$ "! !! !" "" !# !$ "! !! !!
hidden common causes – A challenge of causal discovery – Independence matters rather than uncorrelatedness • Future lines of research – Mixed data with continuous and discrete variables – Multiple datasets – More collaborations with domain experts • Other latent variable models – Latent factors (Shimizu et al., 2009) – latent class (Shimizu et al., 2008) etc. 14 Y. Zeng, S. Shimizu, R. Cai, F. Xie, M. Yamamoto, Z. Hao (2020, arXiv preprint)
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