# E-variable

Addressing the difficulty of incorporating machine learning models into log-optimal e-variable design for conditional independence testing, this work decomposes the error into null enlargement, approximation, and estimation error, explains why exchangeability tests with lower theoretical power can have higher power in practice, explores intermediate null hypotheses between model-X conditional independence and exchangeability to reduce these errors, and provides estimation error bounds accommodating triple robustness results.

Addressing the difficulty of incorporating machine learning models into log-optimal e-variable design for conditional independence testing, this work decomposes the error into null enlargement, approximation, and estimation error, explains why exchangeability tests with lower theoretical power can have higher power in practice, explores intermediate null hypotheses between model-X conditional independence and exchangeability to reduce these errors, and provides estimation error bounds accommodating triple robustness results.

The authors propose sequential resetting procedures built on e-values and test supermartingales that reset the capital to one after each rejection and continue testing, thereby reporting multiple intervals containing null violations; a refinement rule reports only observations after the within-block minimum of the capital. Under independent null e-values the FDR bound is α, while under dependence between null and non-null data the bound is α times a logarithmic factor with anytime validity, demonstrated on LLM watermark detection and ES backtesting.

The authors propose sequential resetting procedures built on e-values and test supermartingales that reset the capital to one after each rejection and continue testing, thereby reporting multiple intervals containing null violations; a refinement rule reports only observations after the within-block minimum of the capital. Under independent null e-values the FDR bound is α, while under dependence between null and non-null data the bound is α times a logarithmic factor with anytime validity, demonstrated on LLM watermark detection and ES backtesting.