Four from four. How good is he really? Bayesian thinking
A new signing scores his first four penalties. Bayesian thinking combines what you believed before with what you've just seen, and shows why a perfect start should nudge your opinion, not replace it.
Beginner Part 1 of Bayesian Football
Contents
The football question
The new signing steps up for his first penalty. Goal. Then his second, third and fourth. Four from four.
Is he a 100% penalty taker? Nobody would say so. But then how good is he? And how much should four kicks change what you thought when he arrived?
The concept
Bayesian thinking has three parts, and football fans use all three without noticing:
- The prior: what you believed before the evidence. Most penalty takers score about three in four, so a new taker probably does too.
- The evidence: what you've just seen. Four from four.
- The posterior: what you believe now, having combined the two.
The prior isn't a guess plucked from the air. It is everything you know about penalty takers in general. The evidence is what you know about this one. The posterior weighs them against each other.
A football example
Say you believe a typical taker scores 75%, and you're about as sure of that as 20 penalties' worth of evidence would make you. Think of that as 15 imaginary goals and 5 imaginary misses.
He scores his first four. Add them to the pile: 19 goals from 24.
$$\text{best estimate} = \frac{15 + 4}{20 + 4} = \frac{19}{24} \approx 79\%$$
In plain football
- 15 and 5 are your belief written as penalties: 75% of 20.
- +4 is his record: four real goals, no misses.
- 24 is everything you now know: 20 penalties' worth of belief plus four real ones.
- So a perfect start moves you from 75% to about 79%. Up, but nowhere near 100%.
Here is how different records move the same starting belief:
| His record | Best estimate | 90% likely between |
|---|---|---|
| None yet | 75.0% | 58.1% and 89.0% |
| 4 from 4 | 79.2% | 64.5% and 91.0% |
| 0 from 2 | 68.2% | 51.3% and 83.2% |
| 9 from 10 | 80.0% | 67.1% and 90.6% |
| 45 from 50 | 85.7% | 78.3% and 91.9% |
Two misses from two drops the estimate to 68%, not 0%. Forty-five from fifty takes it to 86%, much closer to his own 90%. The more he takes, the more his own record counts.
A weighted average
There's a simple way to see what's happening. The best estimate is an average of your belief and his record, each weighted by how much evidence it represents:
$$\begin{aligned} &\text{best estimate} \\ &= \frac{20 \times 75\% + 4 \times 100\%}{20 + 4} = 79.2\% \end{aligned}$$
In plain football
- Your belief is worth 20 penalties; his record is worth 4. So your belief gets five times the say.
- After 50 penalties his record is worth more than your belief, and it outweighs it.
How sure you were matters too. Start from the same 75% held only lightly, worth 4 penalties, and four from four takes you to 87.5%. Hold it stubbornly, worth 200 penalties, and you barely move: 75.5%. Neither is wrong in itself. The question is how much you really know about penalty takers before you've seen this one.
Show the mathsBayes' theorem and the Beta prior. Optional.
Write \(\theta\) for his true scoring rate. Bayes' theorem says the belief after the evidence is proportional to the belief before, times how likely the evidence is for each possible \(\theta\):
$$P(\theta \mid \text{data}) \propto P(\text{data} \mid \theta)\; P(\theta)$$
His record of \(s\) goals from \(t\) penalties is a Binomial, so the likelihood is \(\theta^{s}(1-\theta)^{t-s}\). Take the prior to be a Beta distribution, \(\theta^{\alpha-1}(1-\theta)^{\beta-1}\). Multiplying them:
$$\theta^{\alpha + s - 1}\,(1-\theta)^{\beta + t - s - 1}$$
which is another Beta distribution, \(\text{Beta}(\alpha + s,\ \beta + t - s)\). That is why the update is just adding goals to one side and misses to the other. A prior that stays in the same family after an update is called conjugate.
The mean of \(\text{Beta}(a, b)\) is \(a / (a + b)\), which gives the best estimate above. The 90% range comes from the 5th and 95th percentiles of the posterior.
Why it matters
Football is full of small samples. A striker's first month, a keeper's penalty-save record, the league table after three games, a new manager's first five results. In each case the raw number is misleading on its own, and the sensible answer lies somewhere between "what's usually true" and "what we've just seen".
Bayesian thinking makes that compromise explicit. It is the reason good analysts shrink early-season numbers towards the average, and it is the same instinct that stops a manager rebuilding his team after one good week.
Limitations
- The answer depends on the prior. Two people who start from different beliefs will disagree until the evidence is large enough to overrule both. That's honest, not a flaw, but the prior should always be stated.
- It assumes his true rate is fixed. A taker who changes his technique, or an ageing one, really is a different player.
- A penalty isn't only the taker. The goalkeeper and the pressure of the moment matter too.
Try it yourself
Open How good is he really?, set your own belief about typical takers, then enter a player's record. Try the same record with a weak belief and a stubborn one, and watch how far the curve moves.
Further reading
- Seeing Theory: Bayesian inference, Brown University. Interactive: update a belief with coin flips and watch the posterior form.
- Beta distribution: properties, proofs, exercises, StatLect. The distribution used for the prior here, in full.
- Conjugate prior, Wikipedia. Why a Beta prior with Binomial evidence gives a Beta posterior.
- Bayes' theorem, Wikipedia. The formal statement and its history.