Bayes' theorem

In one sentence

Bayes' theorem tells you how to update your belief about an outcome when you see new evidence, weighing your starting view against how strongly the evidence points one way.

How it works

You start with a prior, which is your belief before the new information, often taken from the pre-match price. Then something happens: a team goes a goal up, a player is sent off, a price drifts before kick-off. Bayes asks how much more likely that evidence is if your outcome is going to happen than if it is not.

If the evidence is far more common among eventual winners than eventual losers, your probability rises sharply. If it is almost as common either way, your belief barely moves. The result is the posterior, your updated probability.

Think of it as a detective's rule. A clue matters only in proportion to how much better it fits one suspect than another.

The maths

• P(H): the prior, your probability for the outcome before the evidence.
• P(E given H): how likely the evidence is if the outcome will happen.
• P(E given not H): how likely the evidence is if the outcome will not happen.
• P(H given E): the posterior, your updated probability.

In plain English: new belief equals old belief, scaled by how much better the evidence fits your outcome than the alternative.

Worked betting example

Football, Match Odds. The home team is 1.67 pre-match, which you treat as a 60% prior. They score the first goal after 20 minutes. From your own historical data you estimate (illustrative figures):

• Home teams that go on to win scored first 70% of the time.
• Home teams that go on to draw or lose scored first 30% of the time.

1. Top line: 0.70 × 0.60 = 0.42.
2. Other route to the same evidence: 0.30 × 0.40 = 0.12.
3. Posterior = 0.42 ÷ (0.42 + 0.12) = 0.42 ÷ 0.54 = 0.778, so 77.8%.
4. Fair odds after the goal = 1 ÷ 0.778 = 1.29.

If Betfair shows the home team at 1.40 after the goal, EV per £1 = 0.778 × 1.40 − 1 = 0.089, about 8.9% before commission. After 2% commission on the winnings: 0.778 × 0.40 × 0.98 − 0.222 = 0.083, about 8.3%.

Before trusting that, check when the goal came; the average likelihoods above ignore the minute.

Where it's good

• In-play updating from a pre-match prior as events happen.
• Blending a model with the market: treat the market as a prior and your model output as evidence.
• Early-season ratings, where a team's last-season level is the prior and new results update it.
• Judging how much to trust a tipster or system after a run of results.
• Reading information from drifts and steams before kick-off.

Limitations and pitfalls

• The two likelihoods drive everything. If your "70% of winners scored first" figure is off, the posterior is off.
• Evidence is rarely independent. Scoring first and dominating shots are linked; multiplying them as if separate double-counts.
• A bad prior poisons the result. Using a stale pre-match price after team news has changed it will mislead you.
• Markets update fast. In liquid in-play markets the Bayesian update is often priced in within seconds.
• Base-rate neglect: people jump on dramatic evidence and forget how strong the prior was.
• It is a rule for updating, not a source of edge. You still need better likelihoods than other traders.

How to build it

• Plain Python for single updates; PyMC or scipy.stats for full Bayesian models with distributions instead of single numbers.
• Data: historical outcomes paired with the evidence you want to use (first goal, half-time score, price moves) and pre-match prices as priors.
• Practical tip: estimate likelihoods separately for favourites and underdogs, since the evidence carries different weight for each.

Related methods

• Conditional probability: the building block Bayes rearranges.
• Bayesian updating: applying Bayes repeatedly as data arrives.
• Conjugate priors: shortcuts that keep repeated updates simple.
• Implied probability: a common source for the prior.