Bayesball: Bayesian Integration in Professional Baseball Batters

Abstract

Pitchers in baseball throw the ball with such high velocity and varying movement that batters only have a few hundred milliseconds to estimate whether to swing and how high to swing--contacting the ball too high or too low may produce hit balls that easily result in an out. Even before the pitcher releases the ball, the batter has some belief, or estimated distribution (a 'prior'), of where the ball may land in the zone. Batters will update this prior belief with information from observing the pitch (the 'likelihood') to calculate their final estimate (the 'posterior'). These models of behavior, called Bayesian models within movement science, predict that when players have better prior information, e.g. because they know the upcoming pitch due to 'tipping', that they will rely more on prior information; by contrast if their prior is less informative, e.g. because the pitch is very random as in the case of a knuckleball, they will instead rely more on the observation. Here we test these models using information from more than a million pitches from professional baseball. We find that batters integrate prior information with noisy observations to manage pitch uncertainty. Moreover, as predicted by a Bayesian model, a batter's estimate of where to swing is biased towards the prior when the pitch is tipped and biased towards the likelihood in the case of pitches with high uncertainty. These results demonstrate that Bayesian ideas are relevant well beyond laboratory experiments and matter in the world of sports.