Thomas Bayes’ biography is a strange one for a man who may have contributed more to the problem of precise knowledge than anyone before or since. No one is exactly sure when Bayes was born and there have been conflicting reports as to when he died. The theorem that bears his name was not even discovered until three years after his death. For all the mystery around Bayes’ life, perhaps no one has provided more to the logic of certainty. Bayes‘ Theorem radically alters how we come to think about the probability that something is true, and that general theory could be set to the task of making better football decisions.

According to Thomas Bayes’ Theorem,

p(A|X) =         p(X|A)*p(A)     
          p(X|A)*p(A) + p(X|~A)*p(~A)

If you can figure out what this means, and you can teach it to match officials all over the world, I submit to you, you will get better officiated football matches. Not only that, but the sum total cost of these revolutionary reforms would only include:

The cost of data collection on important match incidents
The cost of training referees in Bayesian thinking
The cost of (minimal) time in extra match preparation for officials

Cheaper than replay systems, and it requires no extra time out of football matches. It’s practically magic, but instead it’s just maths. 

Let’s say you want to know whether a collision in the box is a dive, a penalty, or just incidental contact. Thomas Bayes’ Theorem teaches us to understand the priors, the general population of dives, fouls, and nothings in the normal course of a football season. If you have an accurate accounting of how often these things happen prior to judging the incident you are witnessing, you can then weigh the evidence you have just seen (called posteriors) compared to these priors and make an informed decision as to whether the call you are inclined to make is correct or not.

Football leagues could make stats and video packages for officials similar to the packages teams make for upcoming opponents, but have the stats and video focus on giving officials better data regarding the priors. Those hands that get struck in the box on free kicks? How often are they foul hand balls? When Paul Scholes goes in on a tackle, how many times does he show his studs? How many times does Luis Suarez actually dive compared to embellishing actual fouls? I suggest that there exists a precise range of answers to these questions, and a concerted effort to lodge those answers in officials’ brains could dramatically improve officiating results. 

But how do past incidents effect what actually happen in any one particular moment? Shouldn’t the official call each incident as they see it on the field? These are logical objections, but the objections highlight a myth surrounding how calls get decided and not the reality. There is no such thing as “objective” decision-making. When officials look at a foul, they are calling on comparisons to memory and learned patterns to make their way through the decisions in a match. They do not objectively “see what happens,” but instead take in an image (often a partially vague image at that), store the image (because every relevant bit of data has changed in an instant), filter out irrelevant data (crowd noise, the fact that Sir Alex Ferguson called you fat last week, etc.), compare the data with previous decisions that they have been trained to distinguish, and deliver a judgment swiftly and effectively. This takes a lot of processing power, and we know that this process requires shortcuts to available information that affects our judgement about difficult interpretations. 

Not only do we know that referees already use references to the past to decide the now, we do the same. What else is a “stone cold penalty” but a penalty in which the incident has a near zero likelihood of being either incidental contact or a dive? When there are difficult calls, the co-commentator’s refrain is unfailingly “I’ve seen those given.” 

In other words, match officials, fans in the stands, players protesting on the field, and viewers at home all use priors in judging an officiating decision, they just use them loosely. We know that Luis Suarez dives a lot and Lionel Messi almost never dives. But these reputations, I would guess, lead to over-sampling for each. I bet that Lionel Messi gets a lot more close calls on fouls than Suarez, and I bet that you could literally step on Luis Suarez’s chest and not get a whistle for him. The question here is not whether Messi is “an honest player” and Suarez is a “known diver,” but instead a question of measuring the probability of each statement with more precise probability. What percentage of times does Luis Suarez dive? I don’t know; I haven’t counted. I’m guessing that the officials haven’t either. But I bet match officials carry the prior of getting conned in the box by Suarez in their memory much more vividly then that time he got his foot clipped on the touchline and play innocuously continued. 

One last point: We recognise the difference that an experienced official can make in an important game. This is because we recognize that they have a wider array of memories and experiences that they can draw upon when weighing priors against posteriors to make generally better decisions. Being a match official requires a lot of skills and abilities, but one of them is sorting out tough calls and getting them right with a high degree of accuracy and speed. In proposing that match officials drill on the priors relevant to their next match, I am not suggesting that they carry one more burden in their heads, but instead suggest replacing one of the mental processes that is done already in an approximate, experiential manner with one in which the experiences of all referees can be at least somewhat transmitted to any one given match official, and a non-standardised process can slowly become more routine, more precise, and even more automatic. Thus eliminating the creep of bias in the minds of officials that comes from the imprecise weights and measures of their limited experience and the reputations of he people on the pitch. 

If nothing else, Thomas Bayes’ Theorem teaches us that we needn’t be so holistic in assigning character traits to footballers based upon a few outstanding incidents. A Luis Suarez flop isn’t the only thing that Suarez does anymore than a Robert Huth chest-stamp is defining of his career as a defender. They both have performed millions of physical movements on the football pitch over their careers, and while not condoning either stepping on people’s chests or cheating in order to win penalties, we, as good Bayesian football supporters, can at least judge these acts in proportion to their frequency. While we might have to wait for the rationality revolution to reach the man with the whistle, we can all at least bring a little bit more into the stands and into the pub while we wait.