Modeling Baseball Player Ability with a Nested Dirichlet Distribution

In this paper we introduce the nested Dirichlet probability distribution and propose a method of using it to model Major League Baseball (MLB) player abilities. To do so, we define fourteen distinct outcome types for any typical plate appearance (excluding intentional walks and bunt attempts), and we assume that every player has an underlying fourteen dimensional ability vector, x, where each element represents the probability that the player will experience the corresponding outcome type in any typical plate appearance. We then use the method of maximum likelihood to fit a nested Dirichlet joint prior distribution on x for all MLB batters (excluding pitchers) over the period from 2003-2006.As the nested Dirichlet (like the Dirichlet distribution) is conjugate prior to multinomial data, this model yields a nested Dirichlet posterior distribution for all players as well. We also present extensions to incorporate age effects and year-to-year variance in player underlying abilities to improve the model's predictive power while maintaining a nested Dirichlet posterior leading to surprising new evidence that the underlying abilities of players (not just their statistical performances) are mean-reverting in some sense. We evaluate the posteriors generated by this extended model as a forecasting tool versus future results, showing that the model's accuracy is competitive with popular projection systems, and that the model demonstrates a reasonable estimate of posterior uncertainty. Finally, we discuss further ideas for extending the model as well as some key applications.