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Bayesian Decision Tree Question

Thread starter #1
Hi David, This question is from P1.T2 pg 105.

Sample Problem #1: Suppose there are three types of managers: the underperformers beat the market only 25% of the time, the in-line performers beat the market 50% of the time, and the outperformers beat the market 75% of the time. Our prior belief is that a manager has a 60% probability of being an in-line performer, a 20% chance of being an underperformer, and a 20% chance of being an outperformer. If the manager beats the market two years in a row what should our updated beliefs be?

Can you please help with the solution to above using Decision tree approach (Not using formula). Appreciate it.
 
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David Harper CFA FRM

David Harper CFA FRM
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#2
Hi @GautamN Please see below; my XLS is here https://www.dropbox.com/s/3e3bk4kcx11z2gz/082019-bayes-multiple.xlsx?dl=0

The final column here shows joint probabilities (rather than conditional probabilities) such that the sum of those in the final column must be 100%. Specifically,
  • The joint Prob (OutPerf ∩ Beats Twice) = 20.0% * (75%^2) = 11.25%
  • The joint Prob (In-line ∩ Beats Twice) = 60.0% * (50%^2) = 15.00%
  • The joint Prob (Under-Perform ∩ Beats Twice) = 20.0% * (25%^2) = 1.25%
Then the conditonal Prob (OutPerf | Beat Twice) = 11.25% / (11.25% + 15.00% + 1.25%) = 11.25% / 27.5%= 40.9%. I hope that's helpful!

 
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