Say “Yes” to Bayes

1. Say “Yes” to Bayes A hopefully helpful handoutbyTristan HubschStrayer University

2. The confusion • Whereas the calculation and the result proclaimed by the “vociferous” student in class were correctly answering the question of the homework problem, there was also a serious misunderstanding. • At the point when the “initial” data for part c. were visible at the bottom of the screen and quoting P(win|Olson) = 0.60, someone called out that the answer was not 0.60, but 0.563.

3. Confusion …cont’d • Now, 0.563 indeed is the value of P(Olson|win), but not of P(win|Olson). • Both probabilities can be calculated from the table of joint and marginal probabilities, given in part b., which were visible on screen at the time of the discussion. (Which is why I refered to them and wrote the formula in terms of them on the board.) • Of course, the latter, P(win|Olson), was in fact also given as initial data.

4. Confusion …cont’d • So, whoever was of the opinion, stated by the “vociferous” student, that the text's result for the answer to the question in part c was correct, was indeed correct. • Also, whoever was of the opinion, stated by the “vociferous” student, that the formula I wrote on the board was wrong, was wrong. • So, six of one, half a dozen of the other…

5. Confusion …cont’d • However, I should have recognized this equivocation immediately, and asked you to clearly state which of the two (very different!) statements you endorse… • …but didn’t. • So, each Student on the list will get the 5 points for standing by their conviction, even if it was the second (wrong) one. • In turn, I get to make this presentation.

6. Lawyers, revisited • So, let’s revisit problem 11: • P(A) = 0.40 and P(O) = 0.60.These will soon turn up as “marginal”. • P(w|A) = 0.70, so P(l|A) = 0.30; alsoP(w|O) = 0.60, so P(l|O) = 0.40.These are conditional probabilities, with the lawyer firm as the condition.

7. Lawyers, revisited …cont’d • The joint probabilities are obtained using P(r&F) = P(r|F)·P(F),where “r” stands for the result, and “F” for the lawyer firm in question. • Doggedly substituting “r → w” and then“r → l”, and “F → A” and then “F → O”, provides the table for part b.:

8. P(A) P(O) P(w&A) P(l&A) P(w&O) P(l&O) P(w) P(l) Lawyers, revisited …cont’d All conditional probabilities can now be obtained by turning the previous formula around: P(r|F) = P(r&F) / P(F),

9. joint probability marginal probability = sum of joint probabilities Bayes’ formula Lawyers, revisited …cont’d • P(r|F) = P(r&F)/P(F) = P(F&r)/P(F) = P(F|r)·P(r)/[P(F&r1)+P(F&r2)+…] = P(F|r)·P(r)/[P(F|r1)·P(r1)+P(F&r2)·P(r2)+…] but also • P(F|r) = P(F&r)/P(r) = P(r&F)/P(r) = P(r|F)·P(F)/[P(r&F1)+P(r&F2)+…] = P(r|F)·P(F)/[P(r|F1)·P(F1)+P(r|F2)·P(F2)+…]

10. given inferred Lawyers, revisited …cont’d • P(w|A) = P(w&A)/P(A) • P(w|O) = P(w&O)/P(O) • P(l|A) = P(l&A)/P(A) • P(l|O) = P(l&O)/P(O) =(0.28)/(0.40)=0.70 =(0.36)/(0.60)=0.60 =(0.12)/(0.40)=0.30 =(0.24)/(0.60)=0.40

11. Lawyers, revisited …cont’d Indeed, I had on the board: • P(w|O) = [P(O|w)·P(w)]   /[P(O|w)·P(w)+P(O|l)·P(l)] = P(O&w)/[P(O&w)+P(O&l)] = P(w&O)/P(O) =(0.36)/(0.60)=0.60 …as given; so, it cannot be wrong!

12. Lawyers, revisited …cont’d And, we also have: • P(A|w) = P(A&w)/P(w) • P(O|w) = P(O&w)/P(w) • P(A|l) = P(A&l)/P(l) • P(O|l) = P(O&l)/P(l) =(0.28)/(0.64)=0.438 =(0.36)/(0.64)=0.563 asked =(0.12)/(0.36)=0.333 =(0.24)/(0.36)=0.667

13. Resolution? In summary: • P(O|w) = P(O&w)/P(w) = (0.36)/(0.64) = 0.563was the correct answer to the question in 11c. • P(w|O) = P(w&O)/P(O) = (0.36)/(0.60) = 0.60is a correct recalculation of a given value, from the data listed in part b., for which the more elaborate Bayesian formula was on the board—not wrong; merely answering a different question.

14. Do we have all the ducks in a row now? 