Semi-Parametric Bayesian Estimation of Sparse Multinomial Probabilities with An Application to The Modelling of Bowling Performance in T20I Cricket

We consider modeling bowling performance in Twenty20 international cricket using a semi-parametric Bayesian approach. The bowling performance can be represented as a contingency table and typically yield a sparse contingency table due to cells with small counts and/or zeros. This sparsity is common in Twenty20 international cricket when we have many classification statuses with many levels, even when the sample size is large. Using a Dirichlet process in our proposed model, the multinomial probability vectors are supported on a discrete space, which enables the borrowing of information across data while providing a natural clustering mechanism. Another important feature of the approach is that this borrowing of information also allows the resulting estimators to handle sparsity, a common concern in multinomial data with many categories. The performance of the approach is compared against some of the standard methods available in the literature; James-Stein, empirical Bayes, and Bayesian multinomial regression estimation. To illustrate our modelling strategy, we suggest a simple way to assess the bowling performance of 175 world-class bowlers.

Keywords:James-Stein estimator; Empirical Bayes estimator; Dirichlet process; Multinomial regression; cricket; Sparse data

List of Abbreviations:MLE: Maximum Likelihood Estimator; ML: Maximum Likelihood; JS: James-Stein; EB: Empirical Bayes; MSE: Mean Squared Error; BMR: Bayesian Multinomial Regression; DP: Dirichlet Process; OP: Overall Proportion; ICC: International Cricket Council