volleyball; Markov chain; transition matrix; Markov chain Monte Carlo; Gibbs sampling; multinomial distribution; Bayesian Model
The Brigham Young University Women's Volleyball Team recorded and rated all skills (pass, set, attack, etc.) and recorded rally outcomes (point for BYU, rally continues, point for opponent) for the entire 2006 home volleyball season. Only sequences of events occurring on BYU's side of the net were considered. Events followed one of these general patterns: serve-outcome, pass-set-attack-outcome, or block-dig-set-attack-outcome. These sequences of events were assumed to be first-order Markov chains where the quality of each contact depended only on the quality of the previous contact but not explicitly on contacts further removed in the sequence. We represented these sequences in an extensive matrix of transition probabilities where the elements of the matrix were the probabilities of moving from one state to another. Each row of the count matrix, consisting of the number of times play moved from one transition state to another during the season, was assumed to have a multinomial distribution. A Dirichlet prior was formulated for each row, so posterior estimates of the transition probabilities were then available using Gibbs sampling. The different paths in the transition probability matrix were followed through the possible sequences of events at each step of the MCMC process to compute the posterior probability density that a perfect pass results in a point, a perfect set results in a point, etc. These posterior probability densities are used to address questions about skill performance in BYU Women's Volleyball.