Lesson 12 of 14 · All lessons in this module
- Random variables: PMF, PDF and CDF
- Expected value, variance and standard deviationVideo lesson
- Variance and covariance shortcuts, and the law of large numbers
- Discrete distributions: uniform, binomial and Poisson
- Continuous uniform and normal distributions, and simulating random draws
- Log-normal and logistic distributions, and the moments of key distributions
- Integrated cases: reading probabilities and moments from a return model
- Conditional probability and probability trees
- Conditional expected values and variancesVideo lesson
- Covariance from a joint probability function
- Joint, marginal and conditional distributions as information arrives
- Bayes' formula: updating a probability with new informationVideo lesson
- Bayes in practice: diffuse priors, missing inputs and screening tests
- Integrated cases: a tree with model forecasts and a Bayes update
This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.
Lesson 12 of 14 · 15 min
Bayes' formula: updating a probability with new information
Bayes' formula turns a prior probability into a posterior probability by multiplying it by how much more, or less, likely the new information is when the event is true.
In short
- The prior is your belief before the news; the posterior is your belief after it.
- The likelihood is the probability of seeing the information if the event is true.
- The unconditional probability of the information comes from the total probability rule over all events.
- Posterior = [likelihood ÷ ] × prior, which is the same as the event's joint probability divided by the total.
- If the likelihood is above , the posterior rises above the prior; if it is below, the posterior falls.
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