| In this thesis, we present a hierarchical stochastic model of the game of baseball aimed at improving our understanding of the various sources of uncertainty in this arena as well as how they impact outcomes and optimal decision making. We begin with a probabilistic model of the distribution of player abilities on a granular level based upon a nested Dirichlet distribution, an extension of the Dirichlet and generalized Dirichlet distributions. We extend the model to allow for randomized and deterministic perturbations to the variables over time, accounting for random noise and age effects in player ability over the course of a career. Using historical data (courtesy of Retrosheet), we show that the model's predictions of player results compare favorably to popular commercial forecasting systems and that the model demonstrates a reasonable estimate of posterior uncertainty. We also present surprising new evidence that the underlying abilities of players (not just their statistical performance) are mean-reverting in some sense.;Using this model of player ability, we fit play result outcomes using a linear model which provides insight into the relative influence of the batter, pitcher, and certain environmental factors (such as the home field advantage) in various contexts. We then use these play prediction models to derive transition probabilities for a series of increasingly complex Markov chain representations of a baseball game and develop techniques to predict game outcomes with respect to these models. We also derive win probabilities for teams in these games (as well as distributions over entire seasons), and show that these predictions are close in overall accuracy to those derived from prediction markets, and that our predictions add information to the gambling line, overcoming particular biases of the market. Finally, we discuss how to evaluate context specific decision problems for specific games with respect to a number of in-game strategies such as bunts, intentional walks, and lineup selection. We also show how our model can be used to estimate the value of specific players to specific teams, allowing for a model of player "fit" in baseball, and discuss how this marginal evaluation can be used to evaluate potential trades, free agent acquisitions, and other roster changes, allowing teams to use such methods to optimize roster moves and potentially save millions of dollars. |
