Editing TheStarsAreRight:CarlNote3 (section)

=='''Recovery? (Bayes)'''==

'''CARL'''

“That’s very clever.”  Carl seems impressed.  “I’ve never heard of Mister Bayes.  Not surprising, inasmuch as I have an engineer’s background, but still.  Clever.  And you also – for seeing how to adapt the paradigm to our rather different situation.”  He smiles again.  “Bravo, my friend.”
 

Carl frowns a bit, thoughtful. 
 

“Is there more of this?  This … mathematics of causality?”  He tips his head.  “Could be very useful.  Though I’m having a bit of trouble putting it into concrete, real-world terms.   Let’s see.  The probability of x, given y … say, the probability of me dying, assuming I caught the plague … is equal to … what?  The probability of me having caught the plague, assuming I died, times the likelihood of me dying in general, all divided by the probability of me catching the plague…. Hmm.  Not intuitive, in the particular case, is it?  But interesting.  I’ll have to think about that.”
 

He nods, obviously pleased.  “Thank you!”

'''REDLAND'''

Jack thinks for a moment.
 

"Let me try to give something akin to a 'practical' example: Let's say we're in New York in the financial district (or whereever the heck it is that the Pentheus offices are located). We're sitting at a cafe when we see a peculiar looking man sitting alone at a table toying with his Pentheus ID card. We think nothing more of it, when another man comes in and says, 'Hello, Donal,' and sits down with the man and they start conversing.
 

"Now, we've been looking for John Elwar's friend, Joc. With a name like Donal, this close to Pentheus, we realize that this Donal might actually be Joc. However, we recall that there are actually three Donals, so that reduces the chance to one in three. If we felt reasonably sure that this was Joc, we'd approach him, but if it is one of the other two Donals, it might be dangerous (due to the murders and whatnot)."
 

Carl notices that, as Redland talks, he seems happier than usual. It's a bit odd, because he so rarely seems happy.
 

"Now, from our prior research, we know that there is a ninety percent chance that Joc works for Pentheus. We don't know much about the other Donals, but from our experience with coincidence and the World Soul, we think that each of the non-Joc Donals has a ten percent chance of working for Pentheus. What then is the true probability that the Donal we see before us is actually Joc?
 

"Well, let's use Baye's Theorem! Now, the unconstrained probability that a guy near Pentheus, named Donal, is our man, Joc, we have stated as one in three. So, let's say the probability of J equals one in three. The probability that a man is a Pentheus employee, assuming that the man is Joc, we have stated as nine in ten. So, let's say the probability of P given J is nine in ten. Finally, the unconstrained probability that a man named Donal is a Pentheus employee, is point one, times two (for the two non-Jocs) plus point nine (for Joc) all divided by three (for the three Donals). So, the probability of P is one point one over three.
 

"That's all we needed. We now just need to multiply probability of P given J with the probability of J and divide by the probability of P. This gives us, let's see, one in three times nine in ten divided by one point one over three. Hmmm, the threes cancel and we have point nine over one point one. Let's convert to a percentage. Hmmm. Multiply by ninety. We then have eighty-one over ninety-nine.
 

"So, the probability that our Donal is actually Joc has been calculated to be about eighty-one point eight percent. With that high of a probabililty, the minute his friend leaves we swoop in to talk to the man."
 

He finishes with a flourish.
 

"Well, anyway, with all the assumptions and whatnot, perhaps it's not that practical, but it's still fun! Also theoretically interesting, I think."


'''CARL'''

Carl laughs, delighted.  “That was impressive!  Did you ever lecture, Jack?”

'''REDLAND'''

Jack smiles sadly. "That is, or was, my job. I'm hoping when all of this is over, if my reputation isn't completely in tatters, to get back to it. I suppose if I can't find a position in Britain, I might be able to secure a position at Auburn or NYU. I know some sympathetic people there.
 

"Ah. One last thing on Baye's Theorem. Normally if I were teaching this, I'd motivate it with the following, perhaps counterintuitive example (we economists love results that are counterintuitive). I'll make it brief!:
 

"Suppose you have a diagnostic ritual for the plague that is 99.9% reliable. You know that plague is relatively rare and infects 1 out of every million people. You perform the ritual on Rebecca and she tests positive. Question: How worried should she be?
 

"Answer: Not very. Probability of testing positive equals 1 in 1000. Probability of testing positive given you have the plague is, basically, 1 in 1. Unconstrained probability of having the plague is 1 in 1,000,000. By Baye's: 1 in 1,000,000 times 1 all divided by 1 in 1000 equals 1 in 1000. Thus, despite testing positive for your 99.9% reliable ritual, she only has one-tenth of one percent chance of having the plague. Put another way, your ritual will return roughly 1000 false positives for every true positive."
 

He closes his eyes for a moment.
 

"Well, if you think that probability theory is the key to staving off the Outsiders, I'll probably be able to pull my ..." he reaches under his blankets and pokes at his ribs for a moment, "... rapidly declining weight."
 

"Say, since it doesn't look like I'll be able to walk anytime soon, is there anything you need me to contemplate before we reach port? I'm a bit hazy on what we were doing traveling to England, but I have to imagine it involves some combination of meeting with Henrik's patron, Cecil Becker, trying to figure out what's going on with the Principle of Compassion, or possibly attempting to discover what Detective Hanson was doing in Sandoy and whether or not it was connected to findings there of the Kansas City Flu. I suppose we could also be there to administer a beating to Keynes for his pernicious efforts to destroy the economy of England and restore me to my rightful position at the university ... but that seems less likely."