A lot of the arguments ITT seem based on a misconception of what probability means and how it behaves. I subscribe to the logical probability view that the probability of a proposition is a number assigned to represent the state of information about the proposition. This will coincide with the frequency and propensity views in some cases (you are about to roll a fair die. What do you think the probability of the proposition "The die will land on 1" is? will give the same answer 1/6 under any of those views), but this view allows discussion of things in which the other views really don't make any sense. If I flip a fair coin behind a wall where you can't see it, under the logical probability view, you assign 50% to "the coin is heads" and 50% to "the coin is tails" (because you're told the coin is fair), even though you know the answer is already physically determined and you know it no longer has a physical propensity to come up 50-50.
Another thing that seems commonly mistaken (or ignored) is the probability space. I'm not going to do it justice here, but for people who aren't going to read a technical wiki page, it's enough to basically treat it a set of disjoint propositions, each with a real number 0<=x<=1 attached, whose numbers add to 1 and behave like you'd expect for the logical operations of and, or, not, and if (if what you're if-ing has P>0). The real number attached a proposition is called the probability of the proposition. A probability space is just a mathematical object- for you to take it seriously as a description of something empirical, the propositions and probabilities need to correspond to what's going on- the propositions need to span all possible outcomes, and you need to justify your choice of their associated numbers. With simple cases, people are used to letting the selection process define the space (roll a die and the space becomes {1,2,3,4,5,6}) and using the indifference principle to fill out the numbers that aren't explicitly defined (in the die case, assigning 1/6 to each number). The indifference principle is mangled in many ways, so formulating it in a coherent way is important, namely "Equal states of knowledge about propositions in different problems should be given equal probability assignments." (which, implies the better-known version if and only if you can transform one proposition into another by symmetry/label exchange methods) It's not enough to have some number of propositions of unknown probability and say "we don't know the exact numbers for these yet, so 1/nth each of what's remaining." You have to have equal states of knowledge about each to justify doing that.
This is most obviously mangled in the posts that argue along the line of "choose a day, then choose an awakening, tails>>>>heads". First of all, choosing a day with some probability is something that requires a probability space to have any meaning. So if you just start off with "it is monday", "it is tuesday" as your propositions (to avoid making the explanation to follow any more difficult), that spans the possibilities of when you wake up. But when you go to assign numbers, you can't use the indifference principle because you know monday and tuesday follow different rules, so you have different information about them, so without more argumentation, you'd just be pulling numbers out of your butt with no principle that justifies them. So that line of reasoning doesn't get you to a justified answer.
What's tricky about this problem is that there is no selection process defined by the problem for the awakening you're looking at, and people are so used to using the selection process to populate the space that they just invent a selection process out of thin air. Days were covered above, but the other popular one ITT is sampling by awakenings. Again, this requires a probability space, the indifference principle doesn't apply because your knowledge about how the awakenings come about is different, so you can't justify the 1/3,1/3,1/3 assignment that's implicit in "twice as likely" argument forms. It's a consistent probability space, but so is 1/5, 2/5, 2/5, and nobody thinks that's a justified answer, so you need a reason beyond "it's what the indifference principle would say if it applied here" to argue why 1/3,1/3,1/3 is even better than 1/5,2/5,2/5, much less a justified answer for the whole scenario. I have yet to see one, and I don't think one exists.
So if there's no physical sampling procedure defined, and the indifference principle doesn't apply to the ones pulled out of thin air anyway, are there any other tools to use to justify an answer? Yes. The principle of reflection. Well, more precisely, the slightly restated version of the principle of reflection that goes "Given a probability for a proposition now, you should have the same probability for that proposition later if your information about its likelihood hasn't changed." Which is like the trivial version of the indifference principle, talking about the same proposition in the same problem.
So now, the real problem. You are told the rules and a coin is flipped. Let's recreate my space from post 35 in equivalent but reworded form, based on my future experience from the 5 possible points in time. There are 5 propositions. The first two from before I go to sleep:
a="I haven't slept. I'll go through some stuff and be told the coin is heads"
b="I haven't slept. I'll go through some stuff and be told the coin is tails"
The next 3 from after I go to sleep
c="I will be awakened and have amnesia, be asked my credence, told the coin is heads, and let go" (HM)
d="I will be awakened and have amnesia, be asked my credence, be put back to sleep, be awakened the next day, be asked my credence, told the coin is tails, and let go" (TM)
e="I will be awakened and have amnesia, be asked my credence, told the coin is tails, and be let go." (TT)
each with associated P(a)..P(e)
These are clearly disjoint propositions about my future experience that span all possible outcomes. Because the coin is fair, I assign a and b equal probability 0<P(a)<1/2. Now, in the most general sense, without trying to assign actual numbers yet, I know that P(c),P(d),P(e) are >0 and that P(c)+P(d)+P(e)=1-P(a)-P(b)=1-2P(a). I believe that if it's not a or b, I will be in c with probability P(c)/(1-2P(a)), d with probability P(d)/(1-2P(a)), and e with probability P(e)/(1-2P(a)). Now, I wake up with amnesia. This means that I am in c/d/e and not a/b. So, since this is exactly the same state of knowledge as two sentences ago, by the trivial indifference principle, I must still believe that I am in c with probability P(c)/(1-2P(a)), d with probability P(d)/(1-2P(a)), and e with probability P(e)/(1-2P(a)). This is where the 2/3 crowd makes a critical and obvious mistake. They say "you're twice as likely to be awakened in a tails world (d or e) than in a heads world (c)!!!". This is a probabilistic statement being made without a reference to a probability space. It is true in this space if and only if P(d)+P(e)=2*P(c) and false otherwise.
So if you just decide to go P(c)=P(d)=P(e), ignorning the fact that the indifference principle doesn't apply, then you run into immediate problems. At a/b, you believe that p=1 you'll go through c/d/e and p=.5 that you'll be told the coin is heads at the end. Then, only upon learning that you're in c/d/e- when you knew there was a 100% chance that you would learn you're in c/d/e, your probability is 2/3 tails at the end, violating the trivial indifference principle. Also, at a/b, you believe there's a 50% chance you'll go through heads stuff and a 50% chance you'll go through tails stuff. Again, upon only learning that you're in c/d/e- going through stuff- your probability is 2/3 tails stuff, again violating the trivial indifference principle. Not only is the P(c)=P(d)=P(e) assignment unjustified, it's actively contraindicated.
And P(c)=P(d)+P(e), P(d)=P(e) (aka 1/2,1/4,1/4) works perfectly and obeys the trivial indifference principle in both cases, as well as the actual indifference principle P(d)=P(e) (given tails, P(monday)=P(tuesday)). Those appear to be the unique c/d/e relations for this problem that obey everything. And, in what is exactly 0 coincidence IMO, this corresponds to what you would get if you sampled old experiments with the procedure 1. Choose an experiment (week) at random. 2. Choose a random awakening from that week. 3. See if it ends in H/T. This would give the 1/2,1/4,1/4 distribution of awakenings selected and a 50/50 H/T ratio in experiments (which is the ratio you get if you just 1. Choose an experiment at random. 2. Look at the coin)
The various monday variants with this experiment aren't worth rehashing because I have nothing to add to post 35. PtB's recent variant, coin being flipped on monday night, is worth looking at in a little more detail. I was wrong before- this variant does NOT have the same counting stats as the original when sampled by the method in the previous paragraph. It gives 1/3,1/3,1/3 with 2/3 tails experiments (experiment-look at coin is still 50/50 obviously). Whether PtB wants to believe it or not, and whether anybody else in the world realizes it, these are clearly, demonstrably (completely independent of asking beauty questions, amnesia, credences, etc) not the same experiment. The resulting experiments, sampled in the same way, give different numbers. And I already explained why with a space and with words and I'm still satisfied with it.
