  Consider the problem of modeling hysteresis for finite-state random walks
using higher-order Markov chains. This Letter introduces a Bayesian framework
to determine, from data, the number of prior states of recent history upon
which a trajectory is statistically dependent. The general recommendation is to
use leave-one-out cross validation, using an easily-computable formula that is
provided in closed form. Importantly, Bayes factors using flat model priors are
biased in favor of too-complex a model (more hysteresis) when a large amount of
data is present and the Akaike information criterion (AIC) is biased in favor
of too-sparse a model (less hysteresis) when few data are present.
