"Okay Meritxell, that's twice in a row, go to the Overly Advanced Student Holding Cell and use Message next time."
"And yes, that's a fair point about Ione telling me about her magic library powers in the first place, but all that stuff falls into the category of 'please don't bring in all the other facts you know'."
Keltham writes on the white-wall:
Have:
P(no-book ◁ Conspiracy) = P(no-book & Conspiracy) / P(Conspiracy) = 75%
P(no-book ◁ Ordinary) = P(no-book & Ordinary) / P(Ordinary) = 30%
"This is as far as we can get by applying the definitions of terms we know, and it doesn't obviously let us derive any further interesting facts or quantities."
"Now, what I want to know is the chance that there's a Conspiracy going on or that things are Ordinary, given that I'm now nearly-certain-except-for-insanity-or-stranger-weirdness, by the way your language needs a shorter word for that, that I was told there was no book of cleric spells available in the Ostenso library. We could write the quantity I'm interested in as follows:"
Want:
P(Conspiracy ◁ no-book) / P(Ordinary ◁ no-book)
"Let's talk a moment about the meaning of that term I just wrote down. First, in both the numerator and denominator, we're conditioning on being in a world in which I was told there was no book. Second, starting from inside that world, we narrow it down, in the numerator, to the worlds where there's a conspiracy; and in the denominator, worlds where it's an ordinary library situation and an honest Ione. By dividing these two numbers, we narrow them down into one number, and that one number is a quantity telling me the relative odds of the Conspiracy World and the Ordinary World; it's how many times more probable the Conspiracy world is than the Ordinary world. This single quantity might say, for example, 'Conspiracy is twice as likely as Ordinary' or 'Conspiracy is one-third as likely as Ordinary'."
"Why phrase it that way, instead of just asking how likely Conspiracy is in an absolute sense, after observing no-book? Why not ask for an answer that says, 'If I see no book, a Conspiracy is 20% probable' or some such? Because to get an absolute probability for Conspiracy given no-book, I'd have to consider every other plausible hypothesis that competes with the Governance conspiracy, including, for example, that Cayden Cailean suddenly mind-controlled Ione to answer falsely, in a way that Governance had nothing to do with."
"By dividing the Conspiracy probability and the Ordinary probability, we can ask about the relative chances there, without dragging in all other possible hypotheses involving say Cayden Cailean."
"Mathematically speaking, what further information do I need to derive that quantity I want, from the quantities I have?"
Need:
???