Skewed Left: Bayes and the Hit By Pitch

When home plate umpire Clint Fagan made the decision not to eject Zack Greinke for hitting Miguel Montero with a pitch, and a subsequent decision to eject Ian Kennedy for hitting Greinke with a pitch, he was answering a couple of probability questions that umpires and eventually Major League Baseball will have to face.

It’s not a question of whether the hit by pitch was intentional or not. You’re never able to answer that question. The probability that the act was intentional from the point of view of the umpire/disciplinarian is never 0, even on the most innocuous-looking play, and it’s pretty much never unless Cole Hamels is just begging for a suspension.

After a beaning occurs, an umpire and then the MLB office are forced then to determine how likely it was that tat beaning was intentional. And there is where they have run into trouble historically. It is my view that the umpires tend to be too beholden to their own warnings and may even be trying to legislate to some unwritten rules.

The second belief we’ll get to later, but as to the first, remember, it is perfectly within an umpire’s rights to eject a player with no warning. In fact, the previous pitcher tossed for a hit batsman, Jason Hammel, was ejected with no warning in the episode I documented semi-seriously here. Inversely, just because there has been a warning doesn’t mean there has to be an ejection. In fact, the inning after the brawl on Tuesday night, Mark Ellis was hit by a pitch with no consequence.

The ejection, surprisingly, did not follow the beaning after and immediately after the warning. But at what point in this dance do you think the plunkings escalated to the point of ejection?

Top 5: Cody Ross hit by a pitch
Bottom 6: Yasiel Puig hit by a pitch in the face
Top 7 (first batter): Miguel Montero hit by a pitch in the back
Bottom 7: Zack Greinke hit by a pitch in the shoulder

Sam Miller laid out the uncertainty of such situations in his piece Wednesday. I would argue that the probability that Greinke was throwing at Montero after two inside pitches and one firmly in the back was slightly lower than the probability that Kennedy meant to hit Greinke, but both rose above what has been considered ejection-level in the past.

But the umpires never really know the true probability. The true probability is either 0 or 1. It has to be—each party was either throwing at the other or wasn’t. Yet from the point of view of an observer, the probability is somewhere in between, and that’s where Bayesian statistics comes into play.

In an event that’s impossible to observe with certainty, probabilities from that point of view can be affected by other information you know about the event.

The example we learned in an introductory statistics class was of a blurry person you observe in the distance. To say that any specific person has a 50 percent chance of being a male is ridiculous in the traditional sense. Either he is or she isn’t. Yet from the point of view of the distant observer, there is a 50 percent chance that the person is a male until the observer realizes that the person is wearing long pants.

If 80 percent of men and only 50 percent of women in this neighborhood wear pants, then the probability of this person being a man given that the person is wearing pants is:

Probability(Male|Pants) = ______________P (Pants|Male) * P (Male)_____________
P (Pants|Male) * P (Male) + P (Pants|Female) * P (Female)

= .80*.5/(.80*.5+.50*.5)
= ~61.5%

In looking for examples of this in baseball, I found a 2012 post by Patriot at Walk Like a Sabermetrician in which he determined the probability that Ubaldo Jimenez’s plunking of Troy Tulowitzki in a spring training game was intentional after the two feuded through the media.

Probability(Intent|HBP) = ______________P (HBP|Intent) * P (Intent)________________
P (HBP|Intent) * P (Intent) + P (HBP|No intent) * P (No intent)

Not that umpires would ever be doing these calculations in their head, but Bayes’ theorem is a good way to think about the problem. Since most pitchers have fairly similar chances of hitting a guy when they’re not aiming for flesh, what changes is that initial figure. The likelier it was going into an at-bat that there would be a plunking, the higher the chance it was intentional in retrospect.

It’s important not to let what actually happened cloud your judgment. Of course, when it turns out that the person down the street was a man, you’re going to say it looked like a man all along. But when you were first looking at him, there was only a 61.5 percent chance.

The question of how likely it was that Greinke was throwing at Montero is one that had to be answered while Montero was on his way to the batter’s box. Sure, it looks obvious now that he’d throw at the next guy to set foot in there, but look at what the announcers had to say as Montero actually was stepping to the plate, and throughout that at-bat.

Leading up to Greinke hitting Montero

Vin Scully on KCAL: Nothing at first, nothing after the first two were inside

Steve Berthiaume: Nothing at first, “Greinke having issues locating” after the first two were inside

Leading up to Kennedy hitting Greinke

Scully: Nothing

Berthiaume: Nothing (sideline reporter was doing most of the talking)

Not a “here we go” or a “this ought to be interesting.” Nobody even mentioned it.

This was not the sort of foresight that accompanied what I considered the year’s most obvious intentional plunking (i.e. the probability of intent closest to 1). Here’s the transcript from both broadcasts of the Pirates-Mets game on May 11, the day after Jordany Valdespin admired a home run.

WPIX (NY) feed

Gary Cohen: Valdespin raised some hackles last night when he hit a home run in the ninth inning and with the Mets far behind in the game, paused, admired and flipped the bat, which led to even more conversation today about how do you legislate about that kind of thing if you’re Terry Collins, and Terry has basically said ‘I throw up my hands; I can’t do anything about it.’

Ron Darling: Yeah, daily they have talks with Jordany and what did Terry say, he said ‘he’s a player with flair and sometimes that flair’s going to get him in trouble.’

Cohen: McKenry setting up inside and (HBP happens here) that’s exactly what happens. You flip your bat on a home run and you can expect at some point, you’re going to have that happen.

Root Sports (PIT) feed

Greg Brown: And now Jordany Valdespin. There’s a story today in New York that he stared at that long home run he hit last night in the ninth inning off Jose Contreras. Manager Terry Collins was asked about it. He said ‘I don’t know about now, but 15 years ago, he’d probably get hit today’ in response to this showing him up with the big upper deck home run. Jordany Valdespin. (HBP happens) And look at that. How about that.

To take a highly non-scientific stab at calculating the probability that this was intentional, I reached out to three people who were at the game and employed by neither team. I asked what they would have said if asked what the chance of a HBP was while Valdespin was walking to the plate against Bryan Morris as a pinch-hitter.

They averaged 80 percent, which is right in line with the television dialogue from both sides and Collins’ quotes. So let’s go with that; it’s not like we’ll ever get a better number, and these are all estimates anyway, with true probabilities impossible to know.

How about the chances that he would hit him if he weren’t trying? Morris is so new in the majors that I’d trust the league figure before I’d trust his small sample size, and plate appearances end with a hit batsman with probability 0.0085. That’s a little high, since some of those are intentional, so let’s call it 0.008, and let’s call the probability that he hits him if he is intending to hit him 90 percent. (Shawn Estes was the outlier, but throwing behind a guy should probably be lumped in here too because that can be deemed ejection-worthy.)

Anyway, in the Morris-Valdespin incident, the probability that Morris was throwing intentionally at Valdespin from the point of view of our three observers was:

Probability(Intent|HBP) = ______________P (HBP|Intent) * P (Intent)________________
P (HBP|Intent) * P (Intent) + P (HBP|No intent) * P (No intent)

= .90*.80/(.90*.80 + .008*.20)

= .72/.7216

= a 99.8% chance that Morris was throwing at Valdespin intentionally.

Yet in the Pirates/Mets case, there was no suspension or even much talk of one. Everybody seemed happy. But even if the Pirates felt vindicated and the Mets were sort of cool with it too, you’re still not allowed to throw a baseball at a batter intentionally.

This leads to a bigger question that can play in to what happened in Los Angeles. Is it possible that instead of determining intent, the umpires are actually trying to codify some unwritten rules?

Was Joe West in the Pirates-Mets game lenient because baseball code says he deserved it? And how much did “don’t throw at the head” play into why Greinke was allowed to stay and Kennedy had to go, or was it just the warning?

When you go back and look at the suspensions that have occurred in the last five years, very few have resulted from unwritten rules violations compared to how often we actually hear about said unwritten rules. Here are all the suspensions, pulled from an MLB document, that were for throwing at a batter before the punishments came down for Dodgers-Diamondbacks.

*Showing up pitcher after a home run
**Stealing in a blowout
***Stepping out of the box late

I would clearly put Tuesday’s episode into the retribution category, but it’s not without some attachment to the game’s unwritten rules. Was the umpire saying that by the unwritten rules of the game, Greinke was allowed a free shot for Puig’s facial?

Most pitcher plunking suspensions like we’ll undoubtedly see from Tuesday’s action stem from a battle of “he started it,” where at some point an umpire has to put an end to it. It just seems like sometimes the umpires are ignoring probability, while being overly controlled by the system of warnings and trying to enforce some rules that aren’t really there when they do it.