22 research outputs found
A Kernel Test for Three-Variable Interactions
We introduce kernel nonparametric tests for Lancaster three-variable
interaction and for total independence, using embeddings of signed measures
into a reproducing kernel Hilbert space. The resulting test statistics are
straightforward to compute, and are used in powerful interaction tests, which
are consistent against all alternatives for a large family of reproducing
kernels. We show the Lancaster test to be sensitive to cases where two
independent causes individually have weak influence on a third dependent
variable, but their combined effect has a strong influence. This makes the
Lancaster test especially suited to finding structure in directed graphical
models, where it outperforms competing nonparametric tests in detecting such
V-structures
A Kernel Test for Three-Variable Interactions with Random Processes
We apply a wild bootstrap method to the Lancaster three-variable interaction measure in order to detect factorisation of the joint distribution on three variables forming a stationary random process, for which the existing permutation bootstrap method fails. As in the i.i.d. case, the Lancaster test is found to outperform existing tests in cases for which two independent variables individually have a weak influence on a third, but that when considered jointly the influence is strong. The main contributions of this paper are twofold: first, we prove that the Lancaster statistic satisfies the conditions required to estimate the quantiles of the null distribution using the wild bootstrap; second, the manner in which this is proved is novel, simpler than existing methods, and can further be applied to other statistics
Large-Scale Kernel Methods for Independence Testing
Representations of probability measures in reproducing kernel Hilbert spaces
provide a flexible framework for fully nonparametric hypothesis tests of
independence, which can capture any type of departure from independence,
including nonlinear associations and multivariate interactions. However, these
approaches come with an at least quadratic computational cost in the number of
observations, which can be prohibitive in many applications. Arguably, it is
exactly in such large-scale datasets that capturing any type of dependence is
of interest, so striking a favourable tradeoff between computational efficiency
and test performance for kernel independence tests would have a direct impact
on their applicability in practice. In this contribution, we provide an
extensive study of the use of large-scale kernel approximations in the context
of independence testing, contrasting block-based, Nystrom and random Fourier
feature approaches. Through a variety of synthetic data experiments, it is
demonstrated that our novel large scale methods give comparable performance
with existing methods whilst using significantly less computation time and
memory.Comment: 29 pages, 6 figure
A low variance consistent test of relative dependency
We describe a novel non-parametric statistical hypothesis test of relative
dependence between a source variable and two candidate target variables. Such a
test enables us to determine whether one source variable is significantly more
dependent on a first target variable or a second. Dependence is measured via
the Hilbert-Schmidt Independence Criterion (HSIC), resulting in a pair of
empirical dependence measures (source-target 1, source-target 2). We test
whether the first dependence measure is significantly larger than the second.
Modeling the covariance between these HSIC statistics leads to a provably more
powerful test than the construction of independent HSIC statistics by
sub-sampling. The resulting test is consistent and unbiased, and (being based
on U-statistics) has favorable convergence properties. The test can be computed
in quadratic time, matching the computational complexity of standard empirical
HSIC estimators. The effectiveness of the test is demonstrated on several
real-world problems: we identify language groups from a multilingual corpus,
and we prove that tumor location is more dependent on gene expression than
chromosomal imbalances. Source code is available for download at
https://github.com/wbounliphone/reldep.Comment: International Conference on Machine Learning, Jul 2015, Lille, Franc
Feature-to-feature regression for a two-step conditional independence test
The algorithms for causal discovery and more broadly for learning the structure of graphical models require well calibrated and consistent conditional independence (CI) tests. We revisit the CI tests which are based on two-step procedures and involve regression with subsequent (unconditional) independence test (RESIT) on regression residuals and investigate the assumptions under which these tests operate. In particular, we demonstrate that when going beyond simple functional relationships with additive noise, such tests can lead to an inflated number of false discoveries. We study the relationship of these tests with those based on dependence measures using reproducing kernel Hilbert spaces (RKHS) and propose an extension of RESIT which uses RKHS-valued regression. The resulting test inherits the simple two-step testing procedure of RESIT, while giving correct Type I control and competitive power. When used as a component of the PC algorithm, the proposed test is more robust to the case where hidden variables induce a switching behaviour in the associations present in the data