Estimation of Causal Direction in the Presence of Latent Confounders Using a Bayesian LiNGAM Mixture Model Naoki Tanaka, Shohei Shimizu, Takashi Washio The Institute of Scientific and Industrial Research, Osaka University Outline 1. 2. 3. 4. 5. Motivation Background Our Approach Our Model: Bayesian LiNGAM Mixture Simulation Experiments 2 Motivation • Recently, estimation of causal structure attracts much attention in machine learning. – Epidemiology – Genetics Cause Sleep problems • The estimation results can be biased if there are latent confounders. Depression mood Latent confounder → Unobserved variables that have more than one observed child variables. 1 2 Observed variables • We propose a new estimation approach that can solve the problem. 3 Outline 1. 2. 3. 4. 5. Motivation Background Our Approach Our Model: Bayesian LiNGAM Mixture Simulation Experiments 4 LiNGAM（Linear Non-Gaussian Acyclic Model） [Shimizu et al., 2006] • The relations between variables are linear. • Observed variables are generated from a DAG (Directed Acyclic Graphs). 3 x1 1.4 x3 e1 1.4 x2 0.8 x1 0.5 x3 e2 x3 e3 1 1 3 0.5 -0.8 2 • External influences are non-Gaussian. • No latent confounders. → are mutually independent. • LiNGAM is an identifiable causal model. 2 5 A Problem of LiNGAM • Latent confounders make dependent. →The estimation results can be biased. 3 = x1 1.4 x3 e1 x2 0.8 x1 0.5 x3 e2 Patients’ condition mild Medicine A Survival rate x1 e1 ' dependent x2 0.8 x1 e2 ' Patients’ condition serious Medicine A Survival rate 6 LiNGAM with Latent Confounders [Hoyer et al., 2008] • LvLiNGAM (Latent variable LiNGAM) = + <() λ + ：Latent variables ・Independent ・Non-Gaussian λ ：Represent effects of on 7 A Problem in Estimation of LiNGAM with Latent Confounders • Existing methods: • An estimation method using overcomplete ICA. [Hoyer et al., 2008] →Suffers from local optima and requires large sample sizes. • Estimates unconfounded causal relations. [Entner and Hoyer, 2011; Tashiro et al., 2012] →Cannot estimate a causal direction of two observed variables that are affected by latent confounders. • We propose an alternative. – Computationally simpler. – Capable of finding a causal direction in the presence of latent confounders. 8 Outline 1. 2. 3. 4. 5. Motivation Background Our Approach Our Model: Bayesian LiNGAM Mixture Simulation Experiments 9 Basic Idea of Our Approach • Assumption – Continuous latent confounders can be approximated by discrete variables. →LiNGAM with latent confounders reduces to LiNGAM mixture model. [Shimizu et al., 2008] • Estimation – Estimation of LiNGAM mixture. [Mollah et al., 2006] • Also suffers from local optima. – Propose to use Bayesian approach. • Bayesian approach for basic LiNGAM. [Hoyer et al., 2009] 10 LiNGAM Mixture Model [Shimizu et al.,2008] • A data generating model of observed variable within class is = − () + () + () <() Matrix form = () + − mean Class 1 0 Class 2 7 1 1 0.8 2 0.8 2 mean 0 + () + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++ + + + + ++++ + + 6 • Existing estimation methods of LiNGAM mixture model also suffer from local optima.[Mollah et al., 2006] 11 Relation of Latent Variable LiNGAM and LiNGAM Mixture (1) • We assume that continuous latent confounders can be approximated by discrete variables having several values with good precision. – The combination of the discrete values determine which “class” an observation belongs to. → within the same class are mutually independent. →It is simpler than incorporating latent confounders in LiNGAM directly. independent x1 1.4 f 3 e1 x2 0.8 x1 0.5 f 3 e2 3 → constant x1 μ1( c ) e1 x2 0.8 x1 μ(2c ) e2 12 Relation of Latent Variable LiNGAM and LiNGAM Mixture (2) • A simple example – If latent confounders 3 and 4 can be approximated by 0 and 1 … Latent Variable LiNGAM = λ + () + <() LiNGAM Mixture = <() reduces 3 0.7 0.9 0.3 1 0 1 − () + () + () 4 0 1 0.6 2 Class 2 4 1 3 2 0 1 (1) = 0 1 (2) = 0.9 (2) (4) (1) (3) 1 (3) =0.3 1 .9 1.2 0.3 0 .3 .2 1 (4) =0.9 1.2 0.7 0.9 0.3 1 1 0 2 (1) = 0 0.6 2 (2) = 0.6 (3) = 0.7 (4) (1) (3) 22(2) 2 .7 .6 .3 0.7 02 (4) = 1.3 13 Outline 1. 2. 3. 4. 5. Motivation Background Our Approach Our Model: Bayesian LiNGAM Mixture Simulation Experiments 14 Bayesian LiNGAM Mixture Model (1) • The data within class are assumed to be generated by the LiNGAM model. → and , the densities of , have no relation to latent confounders , so they are not different between classes. Although 3 x1 1.4 f 3 e1 21 does not change • changes … x2 0.8 x1 0.5 f 3 e2 Density do not change and () are the same between classes, so we replace and () of the LiNGAM mixture model by and : − () + + () = <() • Then their probability density is () = ( − () − ( − () )) < 15 Bayesian LiNGAM Mixture Model (2) • The probability density of the data within each class is mixed according to some weights. () () | = =1 (: The number of classes) • () : multinomial distribution. • The parameters of the multinomial distribution: Dirichlet distribution – A typical prior for the parameters of the multinomial distribution. – Conjugate prior for multinomial distribution. 16 Compare Three LiNGAM Mixture Models • Select the model with the largest log-marginal likelihood. • There are only three (1 , 2 and 3 ) models between two observed variables because of the assumption of acyclicity. 1 2 3 class class class 1 2 1 2 1 2 17 Log-marginal Likelihood of Our Model • Bayes’ theorem • = 1 , … , P = (| ) (: sample size) • Log-marginal likelihood is calculated as follows: log = log , LiNGAM-mixture Prior distribution • We use Monte Carlo integration to compute the integral. • The assumption of i.i.d. data, , = − =1 =1 − − () < 18 Distribution of • follows a generalized Gaussian distribution with zero means. →Includes Gaussian, Laplace, continuous uniform and many non-Gaussian distributions. – = exp 1 2 Γ( ) – ( ) = | | −( ) 2 Γ(3 ) Γ(1 ) – Γ( ) is the Gamma function. ( ) = 1 = 1 = 2 = 10 19 Prior Distributions and the Number of Classes • Prior distribution – and () ~(, ) – ( ), and 2 ~ − (, ) – can be calculated by using the equation of ( ). • How to select the number of classes. Inv-Gamma(3,3) – Note that ‘true ’ does not exist. ② Selects the best number of classes. (painted in orange) 1 … … 2log 1 0.6 2 0.3 0.8 0.5 3 0.1 0.1 0.2 0.1 In a Dirichlet process mixture model, → ∞ ⇒ → log [Antoniak, 1974] 0.3 ① Selects the best model. (letter in red) 20 Outline 1. 2. 3. 4. 5. Motivation Background Our Approach Our Model: Bayesian LiNGAM Mixture Simulation Experiments 21 Simulation Settings(1) • Generated data using a LiNGAM with latent confounders. [Hoyer et al., 2008] • 100 trials. 3 0.7 4 0.9 -1 0.6 0.8 0.3 (This graph is 2 .) 1 5 1 0.8 2 2 • The distributions of latent variables (1 ，2 ，3 ，4 and 5 ) are randomly selected from the following three non-Gaussian distributions: Laplace distribution Mixture of two Gaussian distribution (symmetric) Mixture of two Gaussian distribution (asymmetric) 22 Simulation Settings(2) • Two methods for comparison: – Pairwise likelihood ratios for estimation of non-Gaussian SEMs [Hyvärinen et al., 2013] →Assumes no latent confounders. – PairwiseLvLiNGAM [Entner et al., 2011] →Finds variable pairs that are not affected by latent confounders and then estimate a causal ordering of one to the other. 23 Simulation Results 1 (1 2 ) 100 The number of correct answers The number of correct answers 100 2 (1 → 2 ) 80 60 40 20 0 50 100 200 Sample size 3 (1 ← 2 ) 100 The number of correct answers True: 80 60 40 20 0 50 100 200 Sample size 100Our method 80 60 40 20 0 Pairwise 50measure PairwiseLv LiNGAM 0(Number of 50 outputs) 50 100 200 Sample size • “(Number of outputs)” is the number of estimation by PairwiseLvLiNGAM. – For the details, Correct answers / Number of outputs 1 2 3 50 64/64 6/12 6/16 100 52/52 7/20 5/24 200 42/42 0/14 2/14 • Our method is most robust against existing latent confounders. 24 Conclusions and Future Work • A challenging problem: Estimation of causal direction in the presence of latent confounders. – Latent confounders violate the assumption of LiNGAM and can bias the estimation results. • Proposed a Bayesian LiNGAM mixture approach. – Capable of finding causal direction in the presence of latent confounders. – Computationally simpler: no iterative estimation in the parameter space. • In this simulation, our method was better than two existing methods. • Future work – Test our method on a wide variety of real datasets. 25 26 Histograms of 20 20 20 10 10 10 0 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 G1, sample size:50 G2, sample size:50 G3, sample size:50 20 20 20 10 10 10 0 0 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 G1, sample size:100 G2, sample size:100 G3, sample size:100 20 20 20 10 10 10 0 0 0 1 2 3 4 5 6 7 8 9 10 G1, sample size:200 1 2 3 4 5 6 7 8 9 10 G2, sample size:200 1 2 3 4 5 6 7 8 9 10 G3, sample size:200 27 Density of a Transformation [Hyvärinen et al., 2001] • e.g.） = (1 , … , ) , = (1 , … , ) • is the density of and is the density of ． – is i.i.d data, so = . Similarly, = • We can rewrite LiNGAM in a matrix form. = + ⇔ = ( − )−1 • = 1 det − −1 = 1 det − −1 ( − )−1 • could be permuted by simultaneous equal row and column permutations to be strictly lower triangular due to the acyclicity assumption. [Bollen, 1989] → ( − )−1 is lower triangular whose diagonal elements are all 1． • A determinant of lower triangular equals the product of its diagonal elements. → |( − )−1 | = 1 28 Gaussian vs. Non-Gaussian 2 Gaussian 2 Non-Gaussian (uniform) (1 → 2 ) 2 1 2 1 (1 ← 2 ) 1 1 29

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