Whether two variables are independent or not, in a general setting. As seen in directed graphical models.

Connection with model selection, in the sense that accepting enough true hypotheses leaves you with a residual independent of the predictors. (TODO: clarify.)

## Tests

If you don't merely want to know whether two things are dependent, but how far apart they are, you may want to estimate a probability metric from data.

### Traditional tests

There are special cases where this is easy, e.g. in binary data we have Chi^2 tests; for Gaussian variables it's the same as correlation, so the problem is simply one of covariance estimates. Generally, likelihood tests can easily give us what is effectively a test of this in estimation problems in exponential families. (c&c Basu's lemma.)

### Copula tests

If we know the copula and variables are monotonically related we know the dependence structure already.

### Information criteria

Information criteria effectively do this. (TODO: clarify.)

### Kernel distribution embedding tests

I'm interested in the nonparametric conditional independence tests of GFTS08, using to kernel tricks, although I don't quite get how you conditionalise them.

RCIT (StZV17) implements an approximate kernel distribution embedding conditional independence test via kernel approximation:

Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales cubicly with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed tests scale linearly with sample size and return accurate p-values much faster than KCIT in the large sample size context. CCD algorithms run with RCIT or RCoT also return graphs at least as accurate as the same algorithms run with KCIT but with large reductions in run time.

## Refs

- SuWh07: Liangjun Su, Halbert White (2007) A consistent characteristic function-based test for conditional independence.
*Journal of Econometrics*, 141(2), 807–834. DOI - GFTS08: Arthur Gretton, Kenji Fukumizu, Choon Hui Teo, Le Song, Bernhard Schölkopf, Alexander J Smola (2008) A Kernel Statistical Test of Independence. In Advances in Neural Information Processing Systems 20: Proceedings of the 2007 Conference. Cambridge, MA: MIT Press
- Tala96: Michel Talagrand (1996) A new look at independence.
*The Annals of Probability*, 24(1), 1–34. - Camp06: Luis M. de Campos (2006) A Scoring Function for Learning Bayesian Networks based on Mutual Information and Conditional Independence Tests.
*Journal of Machine Learning Research*, 7, 2149–2187. - StZV17: Eric V. Strobl, Kun Zhang, Shyam Visweswaran (2017) Approximate Kernel-based Conditional Independence Tests for Fast Non-Parametric Causal Discovery.
*ArXiv:1702.03877 [Stat]*. - Stud16: Milan Studený (2016) Basic facts concerning supermodular functions.
*ArXiv:1612.06599 [Math, Stat]*. - CaRS15: Ben Cassidy, Caroline Rae, Victor Solo (2015) Brain Activity: Connectivity, Sparsity, and Mutual Information.
*IEEE Transactions on Medical Imaging*, 34(4), 846–860. DOI - SzRi09: Gábor J. Székely, Maria L. Rizzo (2009) Brownian distance covariance.
*The Annals of Applied Statistics*, 3(4), 1236–1265. DOI - SSGF12: Dino Sejdinovic, Bharath Sriperumbudur, Arthur Gretton, Kenji Fukumizu (2012) Equivalence of distance-based and RKHS-based statistics in hypothesis testing.
*The Annals of Statistics*, 41(5), 2263–2291. DOI - Lede16: Johannes Lederer (2016) Graphical Models for Discrete and Continuous Data.
*ArXiv:1609.05551 [Math, Stat]*. - SHSF09: Le Song, Jonathan Huang, Alex Smola, Kenji Fukumizu (2009) Hilbert Space Embeddings of Conditional Distributions with Applications to Dynamical Systems. In Proceedings of the 26th Annual International Conference on Machine Learning (pp. 961–968). New York, NY, USA: ACM DOI
- MFSS17: Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, Bernhard Schölkopf (2017) Kernel Mean Embedding of Distributions: A Review and Beyond.
*Foundations and Trends® in Machine Learning*, 10(1–2), 1–141. DOI - ZPJS12: Kun Zhang, Jonas Peters, Dominik Janzing, Bernhard Schölkopf (2012) Kernel-based Conditional Independence Test and Application in Causal Discovery.
*ArXiv:1202.3775 [Cs, Stat]*. - ZFGS16: Qinyi Zhang, Sarah Filippi, Arthur Gretton, Dino Sejdinovic (2016) Large-Scale Kernel Methods for Independence Testing.
*ArXiv:1606.07892 [Stat]*. - SpMe95: Peter Spirtes, Christopher Meek (1995) Learning Bayesian networks with discrete variables from data. In Proceedings of the First International Conference on Knowledge Discovery and Data Mining.
- SzRB07: Gábor J. Székely, Maria L. Rizzo, Nail K. Bakirov (2007) Measuring and testing dependence by correlation of distances.
*The Annals of Statistics*, 35(6), 2769–2794. DOI - EmLM03: Paul Embrechts, Filip Lindskog, Alexander J McNeil (2003) Modelling dependence with copulas and applications to risk management.
*Handbook of Heavy Tailed Distributions in Finance*, 8(329–384), 1. - SFGS12: Bharath K. Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf, Gert R. G. Lanckriet (2012) On the empirical estimation of integral probability metrics.
*Electronic Journal of Statistics*, 6, 1550–1599. DOI - BaSS04: Kunihiro Baba, Ritei Shibata, Masaaki Sibuya (2004) Partial Correlation and Conditional Correlation as Measures of Conditional Independence.
*Australian & New Zealand Journal of Statistics*, 46(4), 657–664. DOI - Stud05: Milan Studený (2005)
*Probabilistic conditional independence structures*. London: Springer - JeKH04: Tony Jebara, Risi Kondor, Andrew Howard (2004) Probability Product Kernels.
*Journal of Machine Learning Research*, 5, 819–844. - Kac59: Mark Kac (1959)
*Statistical independence in probability, analysis and number theory*. Washington, DC: Math. Assoc. of America - YaZS16: Shun Yao, Xianyang Zhang, Xiaofeng Shao (2016) Testing mutual independence in high dimension via distance covariance.
*ArXiv:1609.09380 [Stat]*. - GeMi18: Gery Geenens, Pierre Lafaye de Micheaux (2018) The Hellinger Correlation.
*ArXiv:1810.10276 [Math, Stat]*. - ThSS16: Gian-Andrea Thanei, Nicolai Meinshausen Shah Rajen D., Rajen D. Shah (2016) The xyz algorithm for fast interaction search in high-dimensional data.
*Arxiv*, 20(9), 846–851.