Conditional Independence Is Not (Quite) Pointwise Testable
Abstract
Shah and Peters showed that a conditional independence test with finite-sample or uniformly-controlled level has only trivial power against any alternative. Most practical tests, however, only claim pointwise asymptotic level. There have been incorrect claims in the literature of conditional independence tests with pointwise asymptotic level and consistency against any alternative; whether such a test actually exists has remained open. We resolve this question. Even restricting to hypotheses with a bounded density on compact subsets of Euclidean spaces, for any sequence of (possibly randomized) tests with pointwise asymptotic level α, for every ε > 0 there is a conditionally dependent distribution where the test's limsup power is at most α+ ε. This remains true when the level control is required only over conditionally independent distributions with a continuous density and a uniformly continuous conditional law, and the conditionally dependent distributions have smooth densities. On the other hand, the extreme limitation of trivial power against any alternative for uniform-level tests does not apply to tests with pointwise level. We exhibit a test that, without regularity assumptions, has pointwise asymptotic level α, strictly higher power for all alternatives, and power tending to one for all distributions whose dependence exceeds a chosen scalar threshold.
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BibTeX
@misc{sutherland2026conditional,
title = {{Conditional Independence Is Not (Quite) Pointwise Testable}},
author = {Danica J. Sutherland},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24221},
eprint = {2609.24221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24221},
}
RIS
TY - UNPB
TI - Conditional Independence Is Not (Quite) Pointwise Testable
AU - Danica J. Sutherland
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24221
UR - https://arxiv.org/abs/2609.24221
AB - Shah and Peters showed that a conditional independence test with finite-sample or uniformly-controlled level has only trivial power against any alternative. Most practical tests, however, only claim pointwise asymptotic level. There have been incorrect claims in the literature of conditional independence tests with pointwise asymptotic level and consistency against any alternative; whether such a test actually exists has remained open. We resolve this question. Even restricting to hypotheses with a bounded density on compact subsets of Euclidean spaces, for any sequence of (possibly randomized) tests with pointwise asymptotic level α, for every ε > 0 there is a conditionally dependent distribution where the test's limsup power is at most α+ ε. This remains true when the level control is required only over conditionally independent distributions with a continuous density and a uniformly continuous conditional law, and the conditionally dependent distributions have smooth densities. On the other hand, the extreme limitation of trivial power against any alternative for uniform-level tests does not apply to tests with pointwise level. We exhibit a test that, without regularity assumptions, has pointwise asymptotic level α, strictly higher power for all alternatives, and power tending to one for all distributions whose dependence exceeds a chosen scalar threshold.
ER -