Bayes' Theorem

Bayes' theorem is a rule for updating beliefs when new evidence arrives. You start with a prior probability, what you believed before, and adjust it based on how strongly the new evidence points one way or the other, to get a posterior probability. Good traders and forecasters do this informally all the time: a strong earnings report raises the odds of a rally, a central bank comment lowers the odds of a rate cut. Bayes' theorem makes the process precise and protects against common errors such as ignoring base rates or overreacting to weak signals.

The formula#

P(A | B) = P(B | A) × P(A) / P(B)
P(B) = P(B | A) × P(A) + P(B | not A) × P(not A)

A trading example#

A prediction market example#

A market asks whether a central bank will cut rates at its next meeting. You start with the market price, 40%, as your prior. A key inflation report comes in much lower than expected. Historically, reports like this occurred 60% of the time before cuts and 20% of the time before holds.

P(cut | low report) = 0.60 × 0.40 / (0.60 × 0.40 + 0.20 × 0.60) = 0.24 / 0.36 ≈ 67%

If the market only moves to 55%, your estimate suggests the "yes" side may still be cheap. See Reading Odds as Probabilities.

Why base rates matter#

Bayes' theorem shows that a strong signal can still mean little if the event is rare. A screening pattern that is 90% accurate for a rare event may still produce mostly false alarms. Ignoring the prior is called base rate neglect, one of the most common reasoning errors in trading. See Conditional Probability.

Bayesian thinking habits#

1. Start with a base rate: how often does this kind of setup or event work?
2. Ask how much more likely the evidence is if your idea is right than if it is wrong.
3. Update proportionally: strong evidence moves beliefs a lot; weak evidence moves them a little.
4. Keep updating as new information arrives.
5. Avoid seeing only confirming evidence. See Confirmation Bias.

The likelihood ratio shortcut#

Bayes' theorem can be written in odds form:

posterior odds = prior odds × likelihood ratio
likelihood ratio = P(evidence | true) / P(evidence | false)

In the earnings example, prior odds are 70:30 (2.33), the likelihood ratio is 0.50 / 0.20 = 2.5, so posterior odds are about 5.83, which is a probability of about 85%.

Bayes in quantitative trading#

Bayesian methods are used to combine signals, estimate parameters with limited data, shrink noisy estimates toward sensible priors and build models that update in real time. See Bayesian Statistics and Combining Signals.

Frequently asked questions#

What is Bayes' theorem?#

A formula for updating the probability of a hypothesis when new evidence arrives, combining a prior belief with how likely the evidence is under each possibility.

How is Bayes' theorem used in trading?#

To update the odds of an outcome, such as an earnings beat or a rate cut, as new information arrives, and to avoid overreacting to weak signals.

What is base rate neglect?#

Ignoring how common an event is overall when judging the meaning of a signal, which leads to overestimating the chance of rare events.

Next, learn how uncertain outcomes are described in Random Variables.