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Consider the problem of modeling memory effects in discrete-state randomwalks using higher-order Markov chains. This paper explores cross validationand information criteria as proxies for a model's predictive accuracy. Ourobjective is to select, from data, the number of prior states of recent historyupon which a trajectory is statistically dependent. Through simulations, Ievaluate these criteria in the case where data are drawn from systems withfixed orders of history, noting trends in the relative performance of thecriteria. As a real-world illustrative example of these methods, thismanuscript evaluates the problem of detecting statistical dependencies in shotoutcomes in free throw shooting. Over three NBA seasons analyzed, severalplayers exhibited statistical dependencies in free throw hitting probability ofvarious types - hot handedness, cold handedness, and error correction. For the2013-2014 through 2015-2016 NBA seasons, I detected statistical dependencies in23% of all player-seasons. Focusing on a single player, in two of these threeseasons, LeBron James shot a better percentage after an immediate miss thanotherwise. In those seasons, conditioning on the previous outcome makes for amore predictive model than treating free throw makes as independent. Whenextended to data from the 2016-2017 NBA season specifically for LeBron James, amodel depending on the previous shot (single-step Markovian) does not clearlybeat a model with independent outcomes. An error-correcting variable lengthmodel of two parameters, where James shoots a higher percentage after a missedfree throw than otherwise, is more predictive than either model.

Predictive Bayesian selection of multistep Markov chains, applied to the detection of the hot hand and other statistical dependencies in free throws