# Bayes' Theorem > Bayes' theorem shows how to update a probability when new evidence arrives. Learn the formula, trading and prediction market examples, and base rate errors. Source: https://learn.tradelabsai.com/math/bayes-theorem/ Track: Math and Statistics · Level: Intermediate · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Bayes' Theorem", https://learn.tradelabsai.com/math/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) ``` | Term | Name | Meaning | |---|---|---| | P(A) | Prior | Belief before the evidence | | P(B given A) | Likelihood | How likely the evidence is if A is true | | P(B) | Evidence | How likely the evidence is overall | | P(A given B) | Posterior | Updated belief after the evidence | ## A trading example **Example: Does the signal mean what it seems?** A trader wants to know the chance that a stock beats earnings, given that analysts raised estimates in the final month. - Prior: 70% of companies in this sector beat estimates. P(beat) = 0.70. - When a company beats, estimates were raised beforehand 50% of the time. P(raise | beat) = 0.50. - When a company misses, estimates were raised beforehand 20% of the time. P(raise | miss) = 0.20. P(raise) = 0.50 × 0.70 + 0.20 × 0.30 = 0.35 + 0.06 = 0.41. P(beat | raise) = 0.50 × 0.70 / 0.41 ≈ 85%. Upward revisions raise the probability of a beat from 70% to about 85%. See [Guidance and Earnings Revisions](https://learn.tradelabsai.com/fundamentals/guidance-and-earnings-revisions/). ## 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](https://learn.tradelabsai.com/prediction-markets/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](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/psychology/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](https://learn.tradelabsai.com/math/bayesian-statistics/) and [Combining Signals](https://learn.tradelabsai.com/research/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](https://learn.tradelabsai.com/math/random-variables/). ## Continue learning - Next lesson: [Random Variables](https://learn.tradelabsai.com/math/random-variables/) - Previous lesson: [Conditional Probability](https://learn.tradelabsai.com/math/conditional-probability/) - Related: [Conditional Probability](https://learn.tradelabsai.com/math/conditional-probability/): Conditional probability is the chance of an event given that another has happened. Learn the formula, trading examples, base rates and how to use filters properly. - Related: [Bayesian Statistics](https://learn.tradelabsai.com/math/bayesian-statistics/): Bayesian statistics combines prior beliefs with data to estimate uncertain quantities. Learn priors and posteriors, shrinkage, credible intervals and trading uses. - Related: [Reading Odds as Probabilities](https://learn.tradelabsai.com/prediction-markets/reading-odds-as-probabilities/): Learn to turn prediction market prices into probabilities, adjust for spreads and long shot bias, compare with your own estimate and find value with expected value. - Related: [Confirmation Bias](https://learn.tradelabsai.com/psychology/confirmation-bias/): Confirmation bias makes traders seek evidence that supports their view and ignore what contradicts it. Learn how it shows up and simple habits that counter it. - Related: [Probability for Traders](https://learn.tradelabsai.com/math/probability-for-traders/): Probability is the language of uncertainty in trading. Learn the core rules, independent vs dependent events, odds, and how probability shapes sizing and edge.