Bucky comments on Analysis of a Secret Hitler Scenario

Bucky 23 Aug 2019 10:00 UTC
2 points
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 $P (M a r e k l i b e r a l) = 1 / 2$ 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 $P (M a r e k l i b e r a l | a c c u s e |) = 3 / 4$ .
Odds form helps − 1:1 becomes 2:1 after 1 bit of evidence so $P (M a r e k l i b e r a l | a c c u s e |) = 2 / 3$ .
More formally:
$\frac{P (M a r e k l i b e r a l) | a c c u s e}{P (M a r e k f a s c i s t) | a c c u s e} = \frac{P (a c c u s e | M a r e k l i b e r a l)}{P (a c c u s e | M a r e k f a s c i s t)} . \frac{P (M a r e k l i b e r a l)}{P (M a r e k f a s c i s t)} = \frac{1}{0.5} . \frac{0.5}{0.5} = \frac{2}{1}$