Lesson 6 of 6 · All lessons in this module
  1. Expected value, variance and standard deviationVideo lesson
  2. Conditional probability and probability trees
  3. Conditional expected values and variancesVideo lesson
  4. Bayes' formula: updating a probability with new informationVideo lesson
  5. Bayes in practice: diffuse priors, missing inputs and screening tests
  6. Integrated cases: a tree with model forecasts and a Bayes update

Lesson 6 of 6 · 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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