P(accuse)=37.5% because if I am liberal I only accuse Sam if he is fascist (50% of the time) and if I am fascist I accuse only when I am bold and Sam is liberal (25% of the time).
Imagine 100 games are played. Sam gets accused in 37.5 games. Marek is liberal in 25 of those games.
I had a belief that Marek was Fascist with probability about 50% but I should have already updated to ~88% that he was a fascist as opposed to an unlucky liberal (100% of the time he didn’t draw 3 fascist articles he buried a liberal and is a fascist and half the rest of the time he’s a fascist by my prior).
Using Bayes’ theorem and 23% as the probability of drawing 3 fascist policies, I got 81.3% (and not 88%) as the accurate probability that Marek is fascist.
The first mistake you mention is exactly the mistake I make when I don’t convert to odds form as I mentioned here.
If I start with P(Marekliberal)=1/2 and him accusing gives me 1 bit of evidence (he’s twice as likely to accuse if he’s liberal) then the temptation is to split the uncertainty in half and update incorrectly to P(Marekliberal|accuse|)=3/4 .
Odds form helps − 1:1 becomes 2:1 after 1 bit of evidence so P(Marekliberal|accuse|)=2/3.
You’re definitely right about the 2/3rds. I guess I wrote this up too quickly.
I’m not sure if I agree with your next point. It seems like I have the equality,
P(marek fascist|marek passed a fascist policy)=P(marek saw a liberal policy)+P(marek saw 3 fascists)∗P(marek fascist)
Using the fact that the events are disjoint. Maybe I’m missing an easy application of Bayes though?
I agree with the qualitative analysis and the conclusion but I got different answers when I did the same calculations.
I think the correct probability here is 2⁄3, not 75%.
P(accuse)=37.5% because if I am liberal I only accuse Sam if he is fascist (50% of the time) and if I am fascist I accuse only when I am bold and Sam is liberal (25% of the time).
Imagine 100 games are played. Sam gets accused in 37.5 games. Marek is liberal in 25 of those games.
Using Bayes’ theorem and 23% as the probability of drawing 3 fascist policies, I got 81.3% (and not 88%) as the accurate probability that Marek is fascist.
The first mistake you mention is exactly the mistake I make when I don’t convert to odds form as I mentioned here.
If I start with P(Marekliberal)=1/2 and him accusing gives me 1 bit of evidence (he’s twice as likely to accuse if he’s liberal) then the temptation is to split the uncertainty in half and update incorrectly to P(Marekliberal|accuse|)=3/4 .
Odds form helps − 1:1 becomes 2:1 after 1 bit of evidence so P(Marekliberal|accuse|)=2/3.
More formally:
You can improve your tex formatting by putting your text in \text{}
You’re definitely right about the 2/3rds. I guess I wrote this up too quickly.
I’m not sure if I agree with your next point. It seems like I have the equality, P(marek fascist|marek passed a fascist policy)=P(marek saw a liberal policy)+P(marek saw 3 fascists)∗P(marek fascist) Using the fact that the events are disjoint. Maybe I’m missing an easy application of Bayes though?