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

## TL;DR

This paper develops a Bayesian method for selecting the order of multistep Markov chains to model dependencies in sequential data, demonstrated through analyzing free throw shooting in NBA players to detect hot hand effects.

## Contribution

It introduces a predictive Bayesian approach for model selection of Markov chain order, applied to real-world sports data to identify statistical dependencies in free throw outcomes.

## Key findings

- Detected statistical dependencies in 23% of NBA player-seasons.
- LeBron James shows improved free throw percentage after a miss in some seasons.
- A variable length model with error correction outperforms simpler models.

## Abstract

Consider the problem of modeling memory effects in discrete-state random walks using higher-order Markov chains. This paper explores cross validation and information criteria as proxies for a model's predictive accuracy. Our objective is to select, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. Through simulations, I evaluate these criteria in the case where data are drawn from systems with fixed orders of history, noting trends in the relative performance of the criteria. As a real-world illustrative example of these methods, this manuscript evaluates the problem of detecting statistical dependencies in shot outcomes in free throw shooting. Over three NBA seasons analyzed, several players exhibited statistical dependencies in free throw hitting probability of various types - hot handedness, cold handedness, and error correction. For the 2013-2014 through 2015-2016 NBA seasons, I detected statistical dependencies in 23% of all player-seasons. Focusing on a single player, in two of these three seasons, LeBron James shot a better percentage after an immediate miss than otherwise. In those seasons, conditioning on the previous outcome makes for a more predictive model than treating free throw makes as independent. When extended to data from the 2016-2017 NBA season specifically for LeBron James, a model depending on the previous shot (single-step Markovian) does not clearly beat a model with independent outcomes. An error-correcting variable length model of two parameters, where James shoots a higher percentage after a missed free throw than otherwise, is more predictive than either model.