Information-Geometric Blind Spots of Conformal Test Martingales
Abstract
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.
Related
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BibTeX
@misc{szabadvary2026informationgeometric,
title = {{Information-Geometric Blind Spots of Conformal Test Martingales}},
author = {Johan Hallberg Szabadváry},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23131},
eprint = {2610.23131},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23131},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Information-Geometric Blind Spots of Conformal Test Martingales
AU - Johan Hallberg Szabadváry
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23131
UR - https://arxiv.org/abs/2610.23131
AB - 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) p_S(A(z))/t_S(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.
ER -