Semi-parametric Bayesian inference for multi-season baseball data

Abstract

We analyze complete sequences of successes (hits, walks, and sacrifices) for a group of players from the American and National Leagues, collected over 4 seasons. The goal is to describe how players' performances vary from season to season. In particular, we wish to assess and compare the effect of available occasion-specific covariates over seasons. The data are binary sequences for each player and each season. We model dependence in the binary sequence by an autoregressive logistic model. The model includes lagged terms up to a fixed order. For each player and season we introduce a different set of autologistic regression coefficients, i.e., the regression coefficients are random effects that are specific to each season and player. We use a nonparametric approach to define a random effects distribution. The nonparametric model is defined as a mixture with a Dirichlet process prior for the mixing measure. The described model is justified by a representation theorem for order-$k$ exchangeable sequences. Besides the repeated measurements for each season and player, multiple seasons within a given player define an additional level of repeated measurements. We introduce dependence at this level of repeated measurements by relating the season-specific random effects vectors in an autoregressive fashion. We ultimately conclude that while some covariates like the ERA of the opposing pitcher are always relevant, others like an indicator for the game being into the seventh inning may be significant only for certain seasons, and some others, like the score of the game, can safely be ignored.