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June  2020, 2(2): 155-172. doi: 10.3934/fods.2020009

A Bayesian nonparametric test for conditional independence

Onur Teymur and  Sarah Filippi

Department of Mathematics, Imperial College London, UK

Published  July 2020

Fund Project: Supported by EPSRC grant EP/R013519/1

This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Pólya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type.

Keywords: Bayesian nonparametrics, conditional independence testing.
Mathematics Subject Classification: Primary: 62-08; Secondary: 62G10.
Citation: Onur Teymur, Sarah Filippi. A Bayesian nonparametric test for conditional independence. Foundations of Data Science, 2020, 2 (2) : 155-172. doi: 10.3934/fods.2020009
References:
[1]

J. O. Berger and A. Guglielmi, Bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives, J. Amer. Statist. Assoc., 96 (2001), 174-184.  doi: 10.1198/016214501750333045.  Google Scholar

[2]

W. Bergsma, Testing conditional independence for continuous random variables, Report Eurandom, 2004. Google Scholar

[3]

T. B. Berrett, Y. Wang, R. F. Barber and R. J. Samworth, The conditional permutation test for independence while controlling for confounders, J. R. Stat. Soc. B, 82 (2020), 175-197.  doi: 10.1111/rssb.12340.  Google Scholar

[4]

E. Candès, Y. Fan, L. Janson and J. Lv, Panning for gold: Model-X knockoffs for high dimensional controlled variable selection, J. R. Stat. Soc. Ser. B. Stat. Methodol., 80 (2018), 551-577.  doi: 10.1111/rssb.12265.  Google Scholar

[5]

G. Doran, K. Muandet, K. Zhang and B. Schölkopf, A permutation-based kernel conditional independence test, Proc. 30th Conf. UAI, 132–141. Google Scholar

[6]

M. Escobar and M. West, Bayesian density estimation and inference using mixtures, J. Amer. Statist. Assoc., 90 (1995), 577-588.  doi: 10.1080/01621459.1995.10476550.  Google Scholar

[7]

S. Filippi and C. Holmes, A Bayesian nonparametric approach to testing for dependence between random variables, Bayesian Anal., 12 (2017), 919-938.  doi: 10.1214/16-BA1027.  Google Scholar

[8]

R. Fisher, The distribution of the partial correlation coefficient, Metron, 3 (1924), 329-332.   Google Scholar

[9]

K. Fukumizu, A. Gretton, X. Sun and B. Schölkopf, Kernel measures of conditional dependence, Adv. Neural Inf. Process. Syst., 20, 489–496. Google Scholar

[10]

S. Ghosal and A. van der Vaart, Fundamentals of Nonparametric Bayesian Inference, Cambridge Series in Statistical and Probabilistic Mathematics, 44. Cambridge University Press, Cambridge, 2017. doi: 10.1017/9781139029834.  Google Scholar

[11]

J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, Springer-Verlag, New York, 2003.  Google Scholar

[12]

P. Giudici, Bayes factors for zero partial covariances, J. Statist. Plann. Inference, 46 (1995), 161-174.  doi: 10.1016/0378-3758(94)00101-Z.  Google Scholar

[13]

T. E. Hanson, Inference for mixtures of finite Pólya tree models, J. Amer. Statist. Assoc., 101 (2006), 1548-1565.  doi: 10.1198/016214506000000384.  Google Scholar

[14]

T. Hanson and W. O. Johnson, Modeling regression error with a mixture of Pólya trees, J. Amer. Statist. Assoc., 97 (2002), 1020-1033.  doi: 10.1198/016214502388618843.  Google Scholar

[15]

N. Harris and M. Drton, PCalgorithm for nonparanormal graphical models, J. Mach. Learn. Res., 14 (2013), 3365-3383.   Google Scholar

[16]

P. Hoyer, D. Janzing, J. Mooij, J. Peters and B. Schölkopf, Nonlinear causal discovery with additive noise models, Adv. Neural Inf. Process. Syst. 21, 689–696. Google Scholar

[17]

T.-M. Huang, Testing conditional independence using maximal nonlinear conditional correlation, Ann. Statist., 38 (2010), 2047-2091.  doi: 10.1214/09-AOS770.  Google Scholar

[18]

R. E. Kass and A. E. Raftery, Bayes factors, J. Amer. Statist. Assoc., 90 (1995), 773-795.  doi: 10.1080/01621459.1995.10476572.  Google Scholar

[19]

T. Kunihama and D. B. Dunson, Nonparametric Bayes inference on conditional independence, Biometrika, 103 (2016), 35-47.  doi: 10.1093/biomet/asv060.  Google Scholar

[20]

M. Lavine, Some aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 20 (1992), 1222-1235.  doi: 10.1214/aos/1176348767.  Google Scholar

[21]

M. Lavine, More aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 22 (1994), 1161-1176.  doi: 10.1214/aos/1176325623.  Google Scholar

[22]

L. Ma, Adaptive testing of conditional association through recursive mixture modeling, J. Amer. Statist. Assoc., 108 (2013), 1493-1505.  doi: 10.1080/01621459.2013.838899.  Google Scholar

[23]

L. Ma, Recursive partitioning and multi-scale modeling on conditional densities, Electron. J. Stat., 11 (2017), 1297-1325.  doi: 10.1214/17-EJS1254.  Google Scholar

[24] D. J. C. MacKay, Information Theory, Inference and Learning Algorithms, Cambridge University Press, 2003.   Google Scholar
[25]

D. Margaritis, Distribution-free learning of bayesian network structure in continuous domains, Proc. 20th Nat. Conf. Artificial Intel., (2005), 825–830. Google Scholar

[26]

R. D. Mauldin, W. D. Sudderth and S. C. Williams, Pólya trees and random distributions, Ann. Statist., 20 (1992), 1203-1221.  doi: 10.1214/aos/1176348766.  Google Scholar

[27]

S. M. Paddock, Randomized Pólya Trees: Bayesian Nonparametrics for Multivariate Data Analysis, Thesis (Ph.D.)–Duke University. 1999.  Google Scholar

[28] J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511803161.  Google Scholar
[29] J. Peters, D. Janzing and B. Schölkopf, Elements of Causal Inference: Foundations and Learning Algorithms, MIT Press, Cambridge, MA, 2017.   Google Scholar
[30]

J. Peters, J. Mooij, D. Janzing and B. Schölkopf, Causal discovery with continuous additive noise models, J. Mach. Learn. Res., 15 (2014), 2009-2053.   Google Scholar

[31]

J. Ramsey, A scalable conditional independence test for nonlinear, non-Gaussian data, arXiv: 1401.5031. Google Scholar

[32]

J. Runge, Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information, arXiv: 1709.01447. Google Scholar

[33]

F. Saad and V. Mansinghka, Detecting dependencies in sparse, multivariate databases using probabilistic programming and non-parametric Bayes, Proc. Mach. Learn. Res., 46 (2017), 632-641.   Google Scholar

[34]

R. Shah and J. Peters, The hardness of conditional independence testing and the generalised covariance measure, arXiv: 1804.07203. Google Scholar

[35]

P. Spirtes and C. Glymour, An algorithm for fast recovery of sparse causal graphs, Soc. Sci. Comput. Rev., 9 (1991), 62-72.  doi: 10.1177/089443939100900106.  Google Scholar

[36]

E. Strobl, K. Zhang and S. Visweswaran, Approximate kernel-based conditional independence tests for fast non-parametric causal discovery, J. Causal Inference, (2019), 20180017. doi: 10.1515/jci-2018-0017.  Google Scholar

[37]

L. Su and H. White, A consistent characteristic function-based test for conditional independence, J. Econom., 141 (2007), 807-834.  doi: 10.1016/j.jeconom.2006.11.006.  Google Scholar

[38]

L. Su and H. White, A nonparametric Hellinger metric test for conditional independence, Econom. Theory, 24 (2008), 829-864.  doi: 10.1017/S0266466608080341.  Google Scholar

[39]

W. H. Wong and L. Ma, Optional Pólya tree and Bayesian inference, Ann. Statist., 38 (2010), 1433-1459.  doi: 10.1214/09-AOS755.  Google Scholar

[40]

Q. Zhang, S. Filippi, S. Flaxman and D. Sejdinovic, Feature-to-feature regression for a two-step conditional independence test, Proc. 33rd Conf. UAI, 2017. Google Scholar

[41]

K. Zhang, J. Peters, D. Janzing and B. Schölkopf, Kernel-based conditional independence test and application in causal discovery, arXiv: 1202.3775. Google Scholar

[42]

J. Zhang, L. Yang and X. Wu, Pólya tree priors and their estimation with multi-group data, Stat. Pap., 60 (2019), 499-525.  doi: 10.1007/s00362-016-0852-x.  Google Scholar

show all references

References:
[1]

J. O. Berger and A. Guglielmi, Bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives, J. Amer. Statist. Assoc., 96 (2001), 174-184.  doi: 10.1198/016214501750333045.  Google Scholar

[2]

W. Bergsma, Testing conditional independence for continuous random variables, Report Eurandom, 2004. Google Scholar

[3]

T. B. Berrett, Y. Wang, R. F. Barber and R. J. Samworth, The conditional permutation test for independence while controlling for confounders, J. R. Stat. Soc. B, 82 (2020), 175-197.  doi: 10.1111/rssb.12340.  Google Scholar

[4]

E. Candès, Y. Fan, L. Janson and J. Lv, Panning for gold: Model-X knockoffs for high dimensional controlled variable selection, J. R. Stat. Soc. Ser. B. Stat. Methodol., 80 (2018), 551-577.  doi: 10.1111/rssb.12265.  Google Scholar

[5]

G. Doran, K. Muandet, K. Zhang and B. Schölkopf, A permutation-based kernel conditional independence test, Proc. 30th Conf. UAI, 132–141. Google Scholar

[6]

M. Escobar and M. West, Bayesian density estimation and inference using mixtures, J. Amer. Statist. Assoc., 90 (1995), 577-588.  doi: 10.1080/01621459.1995.10476550.  Google Scholar

[7]

S. Filippi and C. Holmes, A Bayesian nonparametric approach to testing for dependence between random variables, Bayesian Anal., 12 (2017), 919-938.  doi: 10.1214/16-BA1027.  Google Scholar

[8]

R. Fisher, The distribution of the partial correlation coefficient, Metron, 3 (1924), 329-332.   Google Scholar

[9]

K. Fukumizu, A. Gretton, X. Sun and B. Schölkopf, Kernel measures of conditional dependence, Adv. Neural Inf. Process. Syst., 20, 489–496. Google Scholar

[10]

S. Ghosal and A. van der Vaart, Fundamentals of Nonparametric Bayesian Inference, Cambridge Series in Statistical and Probabilistic Mathematics, 44. Cambridge University Press, Cambridge, 2017. doi: 10.1017/9781139029834.  Google Scholar

[11]

J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, Springer-Verlag, New York, 2003.  Google Scholar

[12]

P. Giudici, Bayes factors for zero partial covariances, J. Statist. Plann. Inference, 46 (1995), 161-174.  doi: 10.1016/0378-3758(94)00101-Z.  Google Scholar

[13]

T. E. Hanson, Inference for mixtures of finite Pólya tree models, J. Amer. Statist. Assoc., 101 (2006), 1548-1565.  doi: 10.1198/016214506000000384.  Google Scholar

[14]

T. Hanson and W. O. Johnson, Modeling regression error with a mixture of Pólya trees, J. Amer. Statist. Assoc., 97 (2002), 1020-1033.  doi: 10.1198/016214502388618843.  Google Scholar

[15]

N. Harris and M. Drton, PCalgorithm for nonparanormal graphical models, J. Mach. Learn. Res., 14 (2013), 3365-3383.   Google Scholar

[16]

P. Hoyer, D. Janzing, J. Mooij, J. Peters and B. Schölkopf, Nonlinear causal discovery with additive noise models, Adv. Neural Inf. Process. Syst. 21, 689–696. Google Scholar

[17]

T.-M. Huang, Testing conditional independence using maximal nonlinear conditional correlation, Ann. Statist., 38 (2010), 2047-2091.  doi: 10.1214/09-AOS770.  Google Scholar

[18]

R. E. Kass and A. E. Raftery, Bayes factors, J. Amer. Statist. Assoc., 90 (1995), 773-795.  doi: 10.1080/01621459.1995.10476572.  Google Scholar

[19]

T. Kunihama and D. B. Dunson, Nonparametric Bayes inference on conditional independence, Biometrika, 103 (2016), 35-47.  doi: 10.1093/biomet/asv060.  Google Scholar

[20]

M. Lavine, Some aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 20 (1992), 1222-1235.  doi: 10.1214/aos/1176348767.  Google Scholar

[21]

M. Lavine, More aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 22 (1994), 1161-1176.  doi: 10.1214/aos/1176325623.  Google Scholar

[22]

L. Ma, Adaptive testing of conditional association through recursive mixture modeling, J. Amer. Statist. Assoc., 108 (2013), 1493-1505.  doi: 10.1080/01621459.2013.838899.  Google Scholar

[23]

L. Ma, Recursive partitioning and multi-scale modeling on conditional densities, Electron. J. Stat., 11 (2017), 1297-1325.  doi: 10.1214/17-EJS1254.  Google Scholar

[24] D. J. C. MacKay, Information Theory, Inference and Learning Algorithms, Cambridge University Press, 2003.   Google Scholar
[25]

D. Margaritis, Distribution-free learning of bayesian network structure in continuous domains, Proc. 20th Nat. Conf. Artificial Intel., (2005), 825–830. Google Scholar

[26]

R. D. Mauldin, W. D. Sudderth and S. C. Williams, Pólya trees and random distributions, Ann. Statist., 20 (1992), 1203-1221.  doi: 10.1214/aos/1176348766.  Google Scholar

[27]

S. M. Paddock, Randomized Pólya Trees: Bayesian Nonparametrics for Multivariate Data Analysis, Thesis (Ph.D.)–Duke University. 1999.  Google Scholar

[28] J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511803161.  Google Scholar
[29] J. Peters, D. Janzing and B. Schölkopf, Elements of Causal Inference: Foundations and Learning Algorithms, MIT Press, Cambridge, MA, 2017.   Google Scholar
[30]

J. Peters, J. Mooij, D. Janzing and B. Schölkopf, Causal discovery with continuous additive noise models, J. Mach. Learn. Res., 15 (2014), 2009-2053.   Google Scholar

[31]

J. Ramsey, A scalable conditional independence test for nonlinear, non-Gaussian data, arXiv: 1401.5031. Google Scholar

[32]

J. Runge, Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information, arXiv: 1709.01447. Google Scholar

[33]

F. Saad and V. Mansinghka, Detecting dependencies in sparse, multivariate databases using probabilistic programming and non-parametric Bayes, Proc. Mach. Learn. Res., 46 (2017), 632-641.   Google Scholar

[34]

R. Shah and J. Peters, The hardness of conditional independence testing and the generalised covariance measure, arXiv: 1804.07203. Google Scholar

[35]

P. Spirtes and C. Glymour, An algorithm for fast recovery of sparse causal graphs, Soc. Sci. Comput. Rev., 9 (1991), 62-72.  doi: 10.1177/089443939100900106.  Google Scholar

[36]

E. Strobl, K. Zhang and S. Visweswaran, Approximate kernel-based conditional independence tests for fast non-parametric causal discovery, J. Causal Inference, (2019), 20180017. doi: 10.1515/jci-2018-0017.  Google Scholar

[37]

L. Su and H. White, A consistent characteristic function-based test for conditional independence, J. Econom., 141 (2007), 807-834.  doi: 10.1016/j.jeconom.2006.11.006.  Google Scholar

[38]

L. Su and H. White, A nonparametric Hellinger metric test for conditional independence, Econom. Theory, 24 (2008), 829-864.  doi: 10.1017/S0266466608080341.  Google Scholar

[39]

W. H. Wong and L. Ma, Optional Pólya tree and Bayesian inference, Ann. Statist., 38 (2010), 1433-1459.  doi: 10.1214/09-AOS755.  Google Scholar

[40]

Q. Zhang, S. Filippi, S. Flaxman and D. Sejdinovic, Feature-to-feature regression for a two-step conditional independence test, Proc. 33rd Conf. UAI, 2017. Google Scholar

[41]

K. Zhang, J. Peters, D. Janzing and B. Schölkopf, Kernel-based conditional independence test and application in causal discovery, arXiv: 1202.3775. Google Scholar

[42]

J. Zhang, L. Yang and X. Wu, Pólya tree priors and their estimation with multi-group data, Stat. Pap., 60 (2019), 499-525.  doi: 10.1007/s00362-016-0852-x.  Google Scholar

Figure 1.  Construction of a Pólya tree distribution on $ \Omega = [0,1] $. From each set $ C_\ast $, a particle of probability mass passes to the left with (random) probability $ \theta_{\ast0} $ and to the right with probability $ \theta_{\ast1} = 1-\theta_{\ast0} $, with all $ \theta_\ast $ being independently Beta-distributed as described in the main text
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Figure 2.  Pseudocode for the proposed Bayesian nonparametric test for conditional independence
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Figure 3.  Application of the proposed Bayesian testing procedure to four synthetic datasets supported on $ [0,1]^3 $, chosen such that all combinations of unconditional and conditional dependence/independence are represented. The final column gives the ensemble of probabilities of conditional dependence $ p(H_1|W) $ output by the test over 100 repetitions at varying values of data size $ N $, with the blue line representing the median, and the dark and light shaded regions representing the (25, 75)-percentile and (5, 95)-percentile ranges respectively
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Figure 4.  Marginal scatter plots from the CalCOFI Bottle dataset showing the pairwise relationships between $\texttt{Salnty}$, $\texttt{Oxy_µmol.Kg}$ and $\texttt{T_degC}$. The nonlinear nature of the dependences is immediately apparent
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Figure 5.  Example pairwise dependence graphs output by the Bayesian conditional independence test for five variables from the CalCOFI dataset, conditional on $\texttt{T_degC}$, for four different sizes of subsample drawn from the complete dataset. The numbers associated with each edge are the posterior probabilities of conditional dependence $ p(H_1|W^{(N)}) $ and are given to two decimal places; where no edge is shown, this indicates $ p(H_1|W^{(N)})<0.005 $
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Figure 6.  Box-plots giving the output posterior probability of conditional dependence $ p(H_1|W^{(N)}) $ for 100 repetitions of the Bayesian conditional independence test applied to randomly-drawn subsamples of various sizes $ N $ from the CalCOFI dataset. The left-hand plot gives a representative example of a pair of variables conditionally dependent given $\texttt{T_degC}$, while the right-hand plot gives a representative conditionally independent pair
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Figure 3. Top right: 'Slices' from this heatmap with $ \rho = 0.5 $. Bottom: Test outputs for 100 repetitions of the second and third models of Figure 3. Red plots fix $ c = 1 $ (output identical to Figure 3), while the blue plots use the optimising values $ \hat{c} $ from the plot above">Figure 7.  Top left: Heat map of conditional marginal likelihood values for the three constituent models over $ \Omega_X $, $ \Omega_Y $ and $ \Omega_XY $ for the second and third models of Figure 3. Top right: 'Slices' from this heatmap with $ \rho = 0.5 $. Bottom: Test outputs for 100 repetitions of the second and third models of Figure 3. Red plots fix $ c = 1 $ (output identical to Figure 3), while the blue plots use the optimising values $ \hat{c} $ from the plot above
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