Bayesian Decision Tree Question

[GautamN]

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.

 

[David Harper CFA FRM]

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!

 

[GautamN]

Thanks David. This is very helpful.

 

[sailakshmisuresh]

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!

Hi David, I had a doubt in this question. When calculating the joint probability of a manager who is an outperformer and beats the market once , we know that the joint probability is 20%*75% = 0.15. But when we are asked to calculate the probability of a manager who is an outperformer and beats the market two years in a row, shouldn't the probability be (0.15)^2 or ( 0.2^2 * 0.75^2). I am not able to get why we are not taking the square of 0.2 as well. We have 20% of managers who are outperformers every year, then why isn't that also being multiplied twice?


Thank you!

 

[David Harper CFA FRM]

Hi @sailakshmisuresh I love the question because I have to think about it! (I have the advantage of familiarity with this question ...). Let's find the assumption that matches your probability: ( 0.2^2 * 0.75^2). I think such an assumption would be. Being fanciful, let's say that the manager has a suit called "outperformer" and another suit called "blue suit" such that, on any given day, the manager has a 20% probability of wearing the "outperformer suit" but an 80% probability of wearing his/her blue suit. Naturally, suit wearing is independent from day to day (aka, not autocorrelated). Further, when the manager wears the outperformer suit, there is 20% conditional probability (s)he will outperform. Now, in this fanciful setup, the joint probability that the manager both wears the outperforming suit and beats the market is 20%*75%, and the probability (s)he this happens two days in a row is (20%*75%)^2. Why the difference? Because in this fanciful setup, on each new independent day, the it is a random conditional variable as to whether the manager dons the outpeformer suit.

But the actual question assumes that a manager is one of three types. Being an outperformer is a status not a temporary or random feature. If (s)he is an outpeformer, then such status is permanent (at least for this exercise). Put another way, rather than (20%*75%)^2 = (20%*75%)*(20%*75%), we effectively have here (20%*75%)*(100%*75%) because: if the manager was an outperformer on the first day, then it's 100% true that (s)he is also an outperformer on the second day. That's how I look at this ... hope that's helpful,

 

[sailakshmisuresh]

Thank you so much David! I have clearly understood the difference between the two scenarios.