Carissa's Story
Carissa should have put these students through more stressful things so they wouldn't be so stressed out by tests. Look, one of them is crying. That's a failure of Chelish education having failed to give students something to be more scared of than math -
- unless the problem is that the student is scared of Hell and therefore not solvable that way? But they're not even sending the rejects to Hell!
She entertains the temptation to order everyone to believe that they're not going to be executed for math inadequacy, just so that when this situation inevitably happens anyway they'll know it's their own fault for disobeying orders. Somehow, though, it doesn't seem like that would actually fix the problem.
Carissa has a headband that amounts to Fox's Cunning all the time, and has taken to preparing three or four Owl's Wisdoms a day; it's an unfair advantage but life, and death, and everything else, are unfair.
Keltham wants a rule for calculating a likelihood ratio between the 'same' theory and the 'not same' theory.
If they're in fact the same, then she can add up the lefts from both experiments and the rights from both experiments and get odds of left:right of (N1 + N2 + 1):(M1 + M2 +1). If they're different, then the odds for the first experiment are (N1 + 1):(M1 + 1) and the odds for the second experiment are (N2 + 1):(M2 + 1).
Then to get the propensity p from the left:right ratio you have to do that deeply unpleasant thing with the factorials, which is going to make everything else messy. She digs it out of her notes.
p: M!*N!/(M+N+1)!
p1: M1!N1!/(M1 + N1 + 1)!
p2: M2!N2!/(M2 + N2 + 1)!
pcombo: (M1 + M2)! (N1 + N2)!/ (M1 + M2 + N1 + N1 +1!)
Right, okay, so the odds ratio between the SAME and NOT-SAME theories is just pcombo/p1p2. Which looks hideous, (N1+N2)!(M1+M2)!(M1+N1+1)!(M2+N2+1)!/ (N1+N2+M1+M2+1)!M1!N1!M2!N2!, but conceptually it's not all that hideous.
Presumably if you'd done this experiment you'd want to report the actual data, and the counts of 'left' and 'right', and also your likelihood functions on the actual propensities, and then your likelihood function on SAME: DIFFERENT.
She feels like the third question is getting at something particular, not just as the general principle that if your two data sources are different then you wouldn't want to combine them and treat them as more observations from the same process, but she's not actually sure what in particular. Presumably you could just notice that the dense bits of your likelihood function for each look very different from the dense bits of your likelihood function for the combination, and that if you ever notice that it's a bad sign. - are they supposed to formalize that? It seems surprisingly hard to formalize, but she does have almost two hours to go....
Some of the new students look miserable. To be fair she's not at all sure that without a headband she wouldn't be just as miserable. She's going to - not try to formalize that, she thinks, and instead fold up her notes and hand them in to Keltham and then tilt her head at him invitingly and head to the library.