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Chris Moore, Guy Molyneux, Etan Green, and David Daniels on Bayesian umpires

Kevin Lewis points us to a paper by Etan Green and David Daniels, who conclude that “decisions of [baseball] umpires reflect an accurate, probabilistic, and state-specific understanding of their rational expectations—as well as an ability to integrate those prior beliefs in a manner that approximates Bayes rule.”

This is similar to what was found in an earlier empirical article by Guy Molyneux, and this theoretical treatment of the idea by Chris Moore a few years earlier.

I don’t have anything to add here, except to suggest that these people can now all credit each others’ work in this area when going forward.

P.S. There seems to be some confusion. When I said that these people can all credit each others’ work, I didn’t mean to imply that there had been no references so far. In particular, Green and Daniels in their paper do cite Molyneux already.


  1. a researcher says:

    Just skimmed the Green and Daniels paper and the Molyneux article, it looks like the former (which seems most recent?) does credit the latter (see footnote 16). I clicked the Moore article but I got a broken link.

  2. a researcher says:

    Weird, my browser does load the Moore article now. Not sure what was wrong earlier. Doesn’t look like either of the other articles mention the Moore article.

  3. Jorge Guzman says:

    Whether and to what extent people incorporate priors into their posterior decisions seems a fundamental question in science. I don’t care much about umpires, but I would suspect (and research argues) the same “Bayesian” behavior is what occurs in the statistical discrimination of gender and minorities. Interestingly, this leads us to ask if there are ways to remove Bayesian thinking. Such that, for example, employers that meet a Hispanic person do not assume that individual is of lower ability, even though Hispanics have less education on average.

    I also find it interesting that, in this employment example, being a frequentist with Ho: B=0 is not about a random significance test choice, but instead a normative statement of what the world should be (no discrimination). Bringing that back to umpires, a frequentist approach could highlight this gap as an issue, arguing that umpires should be ‘fair’ and call strikes a strike and balls a ball, independent of accumulated strikes or balls.

    • Keith O'Rourke says:

      This relates, I think, to a more general ban of any trade-off of fairness to gain effectiveness (e.g. accuracy).

      For example, in frequentist statistics, the ban on recovering inter-block information in a randomized block experiments or more generally modeling strata as a random effect instead fixed effect. You do get more accurate assessments of treatment effects but at loss of the (idealized) unbiasedness.

  4. Guy says:

    I don’t think umpires’ Bayesian approach to calling balls and strikes raises any “fairness” issue, in the way that race or sex discrimination does. It is true that this creates an *inconsistency* in the effective strike zone, based on the current ball-strike count (such that a pitch that is called a strike at 3-1 might be called a ball at 1-2). But it’s hard to see how this practice is “unfair” to any group or class. Hitters benefit in some counts, but are at a disadvantage in others (and the same is of course true for pitchers). Most importantly, all players know in advance that the ball/strike decision will be influenced by the count — this is not a surprise to the hitter or the pitcher, or to either team. So the key element of fairness in competitive sports — a level playing field — is maintained.

    Alternatively, imagine that umpires were somehow compelled to ignore the count when calling balls and strikes. How would the game be more “fair?” We know that umpires would make many more mistakes in this scenario, and it would probably result in pitchers’ gaining a greater advantage over hitters, thus reducing run scoring overall. But the appropriate level of run scoring is an aesthetic question, a matter of taste not justice. So while this would certainly change the game, I can’t see how it would make it more fair.

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