Lesson 14 of 14 · All lessons in this module
  1. Random variables: PMF, PDF and CDF
  2. Expected value, variance and standard deviationVideo lesson
  3. Variance and covariance shortcuts, and the law of large numbers
  4. Discrete distributions: uniform, binomial and Poisson
  5. Continuous uniform and normal distributions, and simulating random draws
  6. Log-normal and logistic distributions, and the moments of key distributions
  7. Integrated cases: reading probabilities and moments from a return model
  8. Conditional probability and probability trees
  9. Conditional expected values and variancesVideo lesson
  10. Covariance from a joint probability function
  11. Joint, marginal and conditional distributions as information arrives
  12. Bayes' formula: updating a probability with new informationVideo lesson
  13. Bayes in practice: diffuse priors, missing inputs and screening tests
  14. 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 14 of 14 · 15 min

Integrated cases: a tree with model forecasts and a Bayes update

When the end points of a tree are forecasts from a fitted line, turn each branch value into a model output first, then use conditional expectations to roll the tree back, and let Bayes' formula revise the scenario weights when news arrives.

In short

  • A fitted line Y^=a+bX\hat Y = a + bX gives the expected value of Y given X. Each branch's X must be converted into a Y before you weight anything.
  • Roll back from the leaves: conditional expectation within each scenario, then weight by the scenario probabilities. Joint probabilities = scenario probability × branch probability.
  • Because the model is linear, E(Y∣S)=a+b E(X∣S)E(Y\mid S) = a + b\,E(X\mid S). Plugging an expected X into a non-linear model would not work.
  • Conditional variance of Y given a scenario is b2b^2 times the variance of X in that scenario. The intercept does not matter for spread.
  • News changes the scenario weights through Bayes' formula; the leaves and the model stay the same, and only P(S)P(S) is updated.

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