A Bayesian nonparametric test for conditional independence

Onur Teymur; Sarah Filippi

doi:10.3934/fods.2020009

Previous Article
Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model
FoDS Home
This Issue
Next Article
Hierarchical approximations for data reduction and learning at multiple scales

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.
[2]	W. Bergsma, Testing conditional independence for continuous random variables, Report Eurandom, 2004.
[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.
[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.
[5]	G. Doran, K. Muandet, K. Zhang and B. Schölkopf, A permutation-based kernel conditional independence test, Proc. 30th Conf. UAI, 132–141.
[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.
[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.
[8]	R. Fisher, The distribution of the partial correlation coefficient, Metron, 3 (1924), 329-332.
[9]	K. Fukumizu, A. Gretton, X. Sun and B. Schölkopf, Kernel measures of conditional dependence, Adv. Neural Inf. Process. Syst., 20, 489–496.
[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.
[11]	J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, Springer-Verlag, New York, 2003.
[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.
[13]	T. E. Hanson, Inference for mixtures of finite Pólya tree models, J. Amer. Statist. Assoc., 101 (2006), 1548-1565. doi: 10.1198/016214506000000384.
[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.
[15]	N. Harris and M. Drton, PCalgorithm for nonparanormal graphical models, J. Mach. Learn. Res., 14 (2013), 3365-3383.
[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.
[17]	T.-M. Huang, Testing conditional independence using maximal nonlinear conditional correlation, Ann. Statist., 38 (2010), 2047-2091. doi: 10.1214/09-AOS770.
[18]	R. E. Kass and A. E. Raftery, Bayes factors, J. Amer. Statist. Assoc., 90 (1995), 773-795. doi: 10.1080/01621459.1995.10476572.
[19]	T. Kunihama and D. B. Dunson, Nonparametric Bayes inference on conditional independence, Biometrika, 103 (2016), 35-47. doi: 10.1093/biomet/asv060.
[20]	M. Lavine, Some aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 20 (1992), 1222-1235. doi: 10.1214/aos/1176348767.
[21]	M. Lavine, More aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 22 (1994), 1161-1176. doi: 10.1214/aos/1176325623.
[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.
[23]	L. Ma, Recursive partitioning and multi-scale modeling on conditional densities, Electron. J. Stat., 11 (2017), 1297-1325. doi: 10.1214/17-EJS1254.
[24]	D. J. C. MacKay, Information Theory, Inference and Learning Algorithms, Cambridge University Press, 2003.
[25]	D. Margaritis, Distribution-free learning of bayesian network structure in continuous domains, Proc. 20th Nat. Conf. Artificial Intel., (2005), 825–830.
[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.
[27]	S. M. Paddock, Randomized Pólya Trees: Bayesian Nonparametrics for Multivariate Data Analysis, Thesis (Ph.D.)–Duke University. 1999.
[28]	J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, 2009. doi: 10.1017/CBO9780511803161.
[29]	J. Peters, D. Janzing and B. Schölkopf, Elements of Causal Inference: Foundations and Learning Algorithms, MIT Press, Cambridge, MA, 2017.
[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.
[31]	J. Ramsey, A scalable conditional independence test for nonlinear, non-Gaussian data, arXiv: 1401.5031.
[32]	J. Runge, Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information, arXiv: 1709.01447.
[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.
[34]	R. Shah and J. Peters, The hardness of conditional independence testing and the generalised covariance measure, arXiv: 1804.07203.
[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.
[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.
[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.
[38]	L. Su and H. White, A nonparametric Hellinger metric test for conditional independence, Econom. Theory, 24 (2008), 829-864. doi: 10.1017/S0266466608080341.
[39]	W. H. Wong and L. Ma, Optional Pólya tree and Bayesian inference, Ann. Statist., 38 (2010), 1433-1459. doi: 10.1214/09-AOS755.
[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.
[41]	K. Zhang, J. Peters, D. Janzing and B. Schölkopf, Kernel-based conditional independence test and application in causal discovery, arXiv: 1202.3775.
[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.

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.
[2]	W. Bergsma, Testing conditional independence for continuous random variables, Report Eurandom, 2004.
[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.
[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.
[5]	G. Doran, K. Muandet, K. Zhang and B. Schölkopf, A permutation-based kernel conditional independence test, Proc. 30th Conf. UAI, 132–141.
[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.
[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.
[8]	R. Fisher, The distribution of the partial correlation coefficient, Metron, 3 (1924), 329-332.
[9]	K. Fukumizu, A. Gretton, X. Sun and B. Schölkopf, Kernel measures of conditional dependence, Adv. Neural Inf. Process. Syst., 20, 489–496.
[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.
[11]	J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, Springer-Verlag, New York, 2003.
[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.
[13]	T. E. Hanson, Inference for mixtures of finite Pólya tree models, J. Amer. Statist. Assoc., 101 (2006), 1548-1565. doi: 10.1198/016214506000000384.
[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.
[15]	N. Harris and M. Drton, PCalgorithm for nonparanormal graphical models, J. Mach. Learn. Res., 14 (2013), 3365-3383.
[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.
[17]	T.-M. Huang, Testing conditional independence using maximal nonlinear conditional correlation, Ann. Statist., 38 (2010), 2047-2091. doi: 10.1214/09-AOS770.
[18]	R. E. Kass and A. E. Raftery, Bayes factors, J. Amer. Statist. Assoc., 90 (1995), 773-795. doi: 10.1080/01621459.1995.10476572.
[19]	T. Kunihama and D. B. Dunson, Nonparametric Bayes inference on conditional independence, Biometrika, 103 (2016), 35-47. doi: 10.1093/biomet/asv060.
[20]	M. Lavine, Some aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 20 (1992), 1222-1235. doi: 10.1214/aos/1176348767.
[21]	M. Lavine, More aspects of Pólya tree distributions for statistical modelling, Ann. Statist., 22 (1994), 1161-1176. doi: 10.1214/aos/1176325623.
[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.
[23]	L. Ma, Recursive partitioning and multi-scale modeling on conditional densities, Electron. J. Stat., 11 (2017), 1297-1325. doi: 10.1214/17-EJS1254.
[24]	D. J. C. MacKay, Information Theory, Inference and Learning Algorithms, Cambridge University Press, 2003.
[25]	D. Margaritis, Distribution-free learning of bayesian network structure in continuous domains, Proc. 20th Nat. Conf. Artificial Intel., (2005), 825–830.
[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.
[27]	S. M. Paddock, Randomized Pólya Trees: Bayesian Nonparametrics for Multivariate Data Analysis, Thesis (Ph.D.)–Duke University. 1999.
[28]	J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, 2009. doi: 10.1017/CBO9780511803161.
[29]	J. Peters, D. Janzing and B. Schölkopf, Elements of Causal Inference: Foundations and Learning Algorithms, MIT Press, Cambridge, MA, 2017.
[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.
[31]	J. Ramsey, A scalable conditional independence test for nonlinear, non-Gaussian data, arXiv: 1401.5031.
[32]	J. Runge, Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information, arXiv: 1709.01447.
[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.
[34]	R. Shah and J. Peters, The hardness of conditional independence testing and the generalised covariance measure, arXiv: 1804.07203.
[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.
[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.
[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.
[38]	L. Su and H. White, A nonparametric Hellinger metric test for conditional independence, Econom. Theory, 24 (2008), 829-864. doi: 10.1017/S0266466608080341.
[39]	W. H. Wong and L. Ma, Optional Pólya tree and Bayesian inference, Ann. Statist., 38 (2010), 1433-1459. doi: 10.1214/09-AOS755.
[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.
[41]	K. Zhang, J. Peters, D. Janzing and B. Schölkopf, Kernel-based conditional independence test and application in causal discovery, arXiv: 1202.3775.
[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.

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

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Pseudocode for the proposed Bayesian nonparametric test for conditional independence

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Jörg Schmeling. A notion of independence via moving targets. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 269-280. doi: 10.3934/dcds.2006.15.269
[2]	C. Xiong, J.P. Miller, F. Gao, Y. Yan, J.C. Morris. Testing increasing hazard rate for the progression time of dementia. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 813-821. doi: 10.3934/dcdsb.2004.4.813
[3]	Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[4]	Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[5]	Antoni Buades, Bartomeu Coll, Jose-Luis Lisani, Catalina Sbert. Conditional image diffusion. Inverse Problems and Imaging, 2007, 1 (4) : 593-608. doi: 10.3934/ipi.2007.1.593
[6]	Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
[7]	Philippe Destuynder, Caroline Fabre. Few remarks on the use of Love waves in non destructive testing. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 427-444. doi: 10.3934/dcdss.2016005
[8]	Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics and Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
[9]	Christopher Oballe, Alan Cherne, Dave Boothe, Scott Kerick, Piotr J. Franaszczuk, Vasileios Maroulas. Bayesian topological signal processing. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 797-817. doi: 10.3934/dcdss.2021084
[10]	David Simmons. Conditional measures and conditional expectation; Rohlin's Disintegration Theorem. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2565-2582. doi: 10.3934/dcds.2012.32.2565
[11]	Xiaomin Zhou. A formula of conditional entropy and some applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4063-4075. doi: 10.3934/dcds.2016.36.4063
[12]	Deng Lu, Maria De Iorio, Ajay Jasra, Gary L. Rosner. Bayesian inference for latent chain graphs. Foundations of Data Science, 2020, 2 (1) : 35-54. doi: 10.3934/fods.2020003
[13]	Sahani Pathiraja, Sebastian Reich. Discrete gradients for computational Bayesian inference. Journal of Computational Dynamics, 2019, 6 (2) : 385-400. doi: 10.3934/jcd.2019019
[14]	Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[15]	Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems and Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[16]	Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems and Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030
[17]	Monica Pragliola, Daniela Calvetti, Erkki Somersalo. Overcomplete representation in a hierarchical Bayesian framework. Inverse Problems and Imaging, 2022, 16 (1) : 19-38. doi: 10.3934/ipi.2021039
[18]	Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
[19]	Felipe Cucker, Jiu-Gang Dong. A conditional, collision-avoiding, model for swarming. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1009-1020. doi: 10.3934/dcds.2014.34.1009
[20]	Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences

A Bayesian nonparametric test for conditional independence

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A Bayesian nonparametric test for conditional independence

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors