Validity of permutation-test conditional mutual information for Markov order inference

Ascertain whether permutation-test–based conditional mutual information testing using fully randomized surrogate sequences provides a valid inference that a discrete stochastic process has Markov memory order u, specifically determining if non-rejection of the surrogate-based null at a given u constitutes evidence for memory u rather than merely reflecting insufficient power to reject memoryless (iid) behavior.

Background

The paper evaluates memory estimators for discrete stochastic processes and contrasts the proposed Predictability Gain (PG) method with prior approaches. One such prior method estimates Markov chain order by computing conditional mutual information and using permutation tests that randomize the observed sequence to generate surrogate data. Memory is then defined as the smallest u for which the corresponding p-value exceeds a threshold.

The authors note that because surrogates are fully randomized, the null tested is effectively memoryless (iid), raising doubt about the logical link between failing to reject this null at a given u and concluding that the true process has memory u. Clarifying this issue is important to ensure that permutation-based tests do not conflate evidence of higher-order dependence with limitations in detecting departures from iid behavior.