The problem is that we don’t know what P(T | awake) is (the P(A|C) factor in your notation). That’s the whole question at hand. You are implicitly assuming some value for it in this step.
I am applying a theory of probability that I’m taking as reasonable until I reach a contradiction that shows that it isn’t (proof by contradiction). If P is reasonable, it should give a value to statements like P(T | awake), and that value should follow the normal Bayes rules.
I don’t think I ever injected a specific value that wasn’t already there in the assumptions?
Isn’t your application of Bayes’s rule wrong?
P(T | awake, room 1) = P(room 1 | T, awake) P (T | awake) / P(room 1 | awake)
The second and third factors on the RHS are exactly the values under contention.
No, that’s just Bayes, conditional on another factor, awake (I should have written it last for higher clarity).
Bayes is: P(A|B)=P(B|A)*P(A)/P(B)
Bayes with an extra condition is: P(A|B,C)=P(B|A,C)*P(A|C)/P(B|C)
(or you can define Q(X)=P(X|C) and do standard Bayes with Q(X))
The problem is that we don’t know what P(T | awake) is (the P(A|C) factor in your notation). That’s the whole question at hand. You are implicitly assuming some value for it in this step.
I am applying a theory of probability that I’m taking as reasonable until I reach a contradiction that shows that it isn’t (proof by contradiction). If P is reasonable, it should give a value to statements like P(T | awake), and that value should follow the normal Bayes rules.
I don’t think I ever injected a specific value that wasn’t already there in the assumptions?