Conformal test martingales (CTMs) are e-processes for sequentially testing exchangeability via p-values produced by a conformity function A. However, any distribution shift that preserves the push-forward P∘A−1 leaves the p-value distribution invariant, rendering it invisible to every CTM based on A. We formalise this blind spot through the \emph{A-equivalence class} [P]A and show that, for any target distribution T, there is a unique closest element Q∗∈[P]A in KL divergence---the \emph{A-cryptic I-projection}---with explicit density q∗(z)=t(z)pS(A(z))/tS(A(z)) and a Pythagorean decomposition into detectable and cryptic components. When A is the Neyman-Pearson likelihood ratio, the cryptic component vanishes, connecting our result to optimal e-value design. We demonstrate the theory on high-dimensional image data by empirically constructing this I-projection. By conditionally resampling EMNIST letters to identically match the conformity scores of MNIST digits, we demonstrate that a distribution can be shifted to a disjoint support while leaving the test martingale mathematically blinded.
Figures & tables
Figure 1: The A -cryptic I-projection Q∗ satisfies a Pythagorean identity.
Figure 2: Empirical A -cryptic I-projection. Left: Conformity score densities. By construction, the cryptic projection Q∗ empirically matches the null P . Middle: Sleeper/Stayer martingales. The naive shift to T is rapidly detected, whereas Q∗ perfectly tracks the null hypothesis. Right: Image triplets. The Q∗ samples are pristine letters, demonstrating that a distribution can be shifted to a disjoint support within the equivalence class [P]A without altering the test statistic.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Exact A -cryptic I-projection in the Gaussian setting. Left: T with μT=[5,5] (orange) is distant from P (blue) in feature space, but Q∗ (purple) reweights T to recover P ’s score distribution (middle). The CTM for Q∗ remains non-increasing (right).
Figure 4: Exact A -cryptic I-projection in the Gaussian setting. Left: T with μT=[10,10] (orange) is distant from P (blue) in feature space, but Q∗ (purple) reweights T to recover P ’s score distribution (middle). The CTM for Q∗ remains non-increasing (right).
Figure 5: Exact A -cryptic I-projection in the Gaussian setting. Left: T with μT=[9,−2] (orange) is distant from P (blue) in feature space, but Q∗ (purple) reweights T to recover P ’s score distribution (middle). The CTM for Q∗ remains non-increasing (right).