Hiro Minato points us to this recent article by Guy Molyneux:
Baseball fans have long known, or at least suspected, that umpires call balls and strikes differently as the count changes. At 0-2, it seems that almost any taken pitch that is not right down the middle will be called a ball, while at 3-0 it feels like pitchers invariably get the benefit of the doubt. One of the earliest discoveries made possible by PITCHf/x data was the validation of this perception: Researchers confirmed that the effective size of the strike zone at 0-2 is only about two-thirds as large as in a 3-0 count.
One common explanation offered for this pattern is that umpires don’t want to decide the outcome of a plate appearance. Preferring to let the players play, this argument goes, umpires will only call “strike three” or “ball four” if there is no ambiguity about the call.
But Molyneux has another theory that he claims better fits the data. The theory is that umpires are using Bayesian reasoning. I love it!
The argument goes as follows:
– It’s 3-0. You know and I know and the umpire knows that the pitcher’s probably going to throw a strike. Now a pitch comes in that’s a close call. The “base rate” (in Tversky and Kahneman terms) is that most of those 3-0 pitches are strikes. The Bayesian thing to do is to multiply the likelihood ratio by the ratio of base rates, and the result is that, as the umpire, you should call these close ones as strikes.
– Or, it’s 0-2. We know the pitcher is likely to throw this one away. The base rates is that the pitch is likely to be a ball. Now you take the exact same close call as before—the same “likelihood ratio,” in statistics jargon—but now you multiply it by this new ratio of base rates, and it’s rational to call this one as a ball.
It’s statistical discrimination, baseball-style. Rational in each individual case, with a predictable bias in the aggregate.
This is really a beautiful argument of Molyneux. I’ve never seen it before, but it makes so much sense. And that’s why I say it’s the coolest sports-statistics idea since Miller and Sanjurjo’s work on the hot hand.
P.S. See here for more on Bayesian umpires from Chris Moore in 2009.