MathDB

Pointwise asymptotic conditional-independence testing problem

Fix α∈(0,1)\alpha\in(0,1). Let P\mathcal{P} be a class of distributions of (X,Y,Z)(X,Y,Z) on Euclidean spaces having densities bounded on compact subsets, and let H0={P∈P:X⊥Y∣Z}H_0=\{P\in\mathcal{P}:X\perp Y\mid Z\} and H1=P∖H0H_1=\mathcal{P}\setminus H_0. Does there exist a sequence of, possibly randomized, tests φn\varphi_n based on nn independent observations such that ∀P∈H0,lim sup⁡n→∞EP[φn]≤α,\displaystyle \forall P\in H_0,\quad \limsup_{n\to\infty}\mathbb{E}_P[\varphi_n]\leq\alpha, while ∀P∈H1,lim⁡n→∞EP[φn]=1\displaystyle \forall P\in H_1,\quad \lim_{n\to\infty}\mathbb{E}_P[\varphi_n]=1?

References

Primary source

arXiv

Additional references

hypothesis-testingprobabilitystatistics
Posted · edited

Progress summary

Refreshed
Claimed progress

A September 2026 paper argues that broad guarantees are impossible, while a restricted test can still detect dependence reliably.

The problem concerns the boundary between pointwise and uniform guarantees for conditional-independence tests. Sutherland’s paper claims both a broad impossibility theorem and a positive result under additional restrictions.

Known results

  • Shah and Peters established that unrestricted conditional-independence tests with continuous conditioning variables can be powerless against every alternative under uniform validity.
  • Their Generalised Covariance Measure test achieves nontrivial power when conditional means are sufficiently estimable.
  • Later work records the same impossibility even for bounded variables and extends the restricted positive result to sequential testing.

September 2026 impossibility and positive test

On September 21, 2026, Danica J. Sutherland’s paper Conditional Independence Is Not (Quite) Pointwise Testable reported that uniformly strong pointwise consistency is impossible in a broad setting, while constructing a pointwise-level test with improved power and consistency above a dependence threshold. The impossibility and positive results have different regularity and power scopes; the claims remain unverified here.

Current status (as of September 2026): broad unrestricted guarantees are claimed impossible, while a restricted pointwise test is claimed to work above a dependence threshold; independent verification of the new results is not recorded.

Sources

Solutions 0

No solutions have been posted yet.