The Annals of Applied Statistics

Bayesball: A Bayesian hierarchical model for evaluating fielding in major league baseball

Shane T. Jensen, Kenneth E. Shirley, and Abraham J. Wyner

Full-text: Open access


The use of statistical modeling in baseball has received substantial attention recently in both the media and academic community. We focus on a relatively under-explored topic: the use of statistical models for the analysis of fielding based on high-resolution data consisting of on-field location of batted balls. We combine spatial modeling with a hierarchical Bayesian structure in order to evaluate the performance of individual fielders while sharing information between fielders at each position. We present results across four seasons of MLB data (2002–2005) and compare our approach to other fielding evaluation procedures.

Article information

Ann. Appl. Stat., Volume 3, Number 2 (2009), 491-520.

First available in Project Euclid: 22 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Spatial models Bayesian shrinkage baseball fielding


Jensen, Shane T.; Shirley, Kenneth E.; Wyner, Abraham J. Bayesball: A Bayesian hierarchical model for evaluating fielding in major league baseball. Ann. Appl. Stat. 3 (2009), no. 2, 491--520. doi:10.1214/08-AOAS228.

Export citation


  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • BIS (2007). Baseball info solutions. Available at
  • Dewan, J. (2006). The Fielding Bible. ACTA Sports, Skokie, IL.
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 1 515–533.
  • Gelman, A., Carlin, J., Stern, H. and Rubin, D. (2003). Bayesian Data Analysis, 2nd ed. Chapman & Hall, Boca Raton, FL.
  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transaction on Pattern Analysis and Machine Intelligence 6 721–741.
  • Glickman, M. E. and Stern, H. S. (1998). A state-space model for national football league scores. J. Amer. Statist. Assoc. 93 25–35.
  • Jensen, S. T., Shirley, K. and Wyner, A. J. (2009). Supplement to “Bayesball: A Bayesian hierarchical model for evaluating fielding in major league baseball.” DOI: 10.1214/08-AOAS228SUPP.
  • Kalist, D. E. and Spurr, S. J. (2006). Baseball errors. Journal of Quantitative Analysis in Sports 2 Article 3.
  • Lichtman, M. (2003). Ultimate zone rating. The Baseball Think Factory, March 14, 2003.
  • Pinto, D. (2006). Probabilistic models of range. Baseball Musings, December 11, 2006.
  • Reich, B. J., Hodges, J. S., Carlin, B. P. and Reich, A. M. (2006). A spatial analysis of basketball shot chart data. Amer. Statist. 60 3–12.
  • Thorn, J. and Palmer, P. (1993). Total Baseball. Harper Collins, New York.

Supplemental materials