Bayes' theorem

The question

"A leak says the striker is benched. The price hasn't moved yet. How much should I believe it, and how much is it worth?"

How should team news change your odds? Most bettors either ignore news or jump all the way to certainty. Bayes' theorem sits between those two mistakes: it tells you exactly how far a piece of news should move you, given how reliable the source is and what you believed before.

The idea in one sentence

Start with what you believed, multiply by how much more likely the news is if it's true than if it's false, and that gives your new belief.

The picture

Bayes is easiest with counts rather than percentages. Imagine 1,000 matches where a key striker has a knock, and in each one you hear from the same source.

• The striker actually misses 250 of them (your starting belief, 25%).
• In those 250, the source correctly says "out" 225 times (90%).
• In the other 750, where he plays, the source wrongly says "out" 150 times (20%).

So the source says "out" 225 + 150 = 375 times, and it's right in 225 of them. When you hear "out", the striker is out 225 ÷ 375 = 60% of the time. Not 90%, and not 25%.

The chart below draws those 1,000 matches as a grid of dots. The dots where the source says "out" light up, and the share of lit dots that really are "out" is your new belief.

Worked Betfair example

The Match Odds market has the home side at 2.06, about 48.5%. There's a doubt over their main striker, and you rate it a 25% chance he misses out. With him, you make the home side 50%. Without him, 44%. (Illustrative figures.)

1. Your view before any news. 0.75 × 50% + 0.25 × 44% = 48.5%, a fair price of 2.06. You agree with the market, so there's no bet.
2. The news. An account that follows the club says he's benched. From its track record, it says "out" in 90% of cases when a player really is out, and in 20% of cases when he isn't.
3. The likelihood ratio. 0.90 ÷ 0.20 = 4.5. The news is four and a half times more likely if he's really out.
4. Update in odds form. Your starting odds that he's out are 25 : 75, or 1 : 3 (0.333). Multiply by 4.5 to get 1.5 : 1, which is 1.5 ÷ 2.5 = 60%.
5. Feed that into the match. New home chance = 0.40 × 50% + 0.60 × 44% = 46.4%, a fair price of 2.155.
6. Is laying at 2.06 worth it? Lay £50 at 2.06, a liability of £53. If the home side doesn't win (53.6%) you keep £50, or £49 after 2% commission. Expected value: 0.536 × £49 − 0.464 × £53 = £26.26 − £24.59 = +£1.67.
7. A second source. A local journalist then also says "out". Their record: 80% when out, 10% when fit, a likelihood ratio of 8. Odds go from 1.5 to 1.5 × 8 = 12 : 1, or 92.3%. The home fair price becomes 2.25.

Verdict: one decent source moves you from 25% to 60%, not to certainty. Two independent sources get you above 90%. The value comes from updating faster and more accurately than the market, and it vanishes the moment the line-up is official.

The formula

Bayes' theorem

• H is the hypothesis, "the striker is out".
• E is the evidence, "the source says out".
• P(H) is your belief before the news, called the prior.
• P(E | H) is how often the source says "out" when he really is out.
• P(E | not H) is how often it says "out" when he's actually fit, the false alarm rate.
• P(H | E) is your belief after the news, called the posterior.

In plain English: of all the times you'd hear this news, what share come from the world where it's true? That's the same "given" idea from Lesson 3.2, turned around.

The odds form (the one to remember)

• The left side is your new odds that H is true.
• The first fraction on the right is your old odds.
• The second fraction is the likelihood ratio, LR.

In plain English: new odds = old odds × LR. An LR of 1 means the news is worthless. An LR of 4.5 multiplies your odds by 4.5. Each independent piece of news multiplies again.

From odds back to probability

In plain English: odds of 1.5 : 1 means 1.5 chances for every 1 against, so 1.5 out of 2.5, which is 60%.

Try it

Set the prior to 25%, "says out when out" to 90% and "says out when fit" to 20%, and check you get 60%. Then push the false alarm rate down to 5% and watch how fast the posterior climbs. A source's false alarms matter more than its hits.

Common mistakes

• Taking the source's hit rate as the answer. "They're right 90% of the time" doesn't make it 90%. With a 25% prior and a 20% false alarm rate, it's 60%.
• Ignoring the prior. The same news means much less about a player who is almost always fit. A 5% prior with the same source only gets you to about 19%.
• Counting the same news twice. Five accounts retweeting one leak is one piece of evidence, not five. Only multiply likelihood ratios for sources that are truly independent.
• Forgetting the market is also updating. The price may already have moved on the same news. Your edge is the gap between your posterior and the price you can actually get, which is why closing line value is the honest test.
• Updating on noise. A source with a likelihood ratio near 1, like a pundit's "hunch", should barely move you at all.

Bayes is the backbone of the Bayesian updating used in ratings models, and misjudging your own confidence is one of the reasons good models still lose money.