Sure. So let’s change the bolded text to the clearer convention “Choices ascribed to the reader are probabilistically independent of variables in the setup they don’t know about” (i.e., in this case, the label one asks about is to be considered independent from Tyrrell’s coin flip, further details about how one chooses their question being therefore unimportant).
To put it another way: I was just drawing attention, with that bit, to the [rarely-stated, “obvious”, implicit, etc.] convention in the language of these problems that one cannot gain any information about an “external” fact simply by observing the results of one’s own private “internal” decision-making method (whatever that method may be).
I.e., one cannot reason “Well, I am known to have some psychic affinity, so the very fact that I am described in the setup as choosing to ask about so-and-so can be taken as strong evidence for such-and-such…”. There’ll be none of that. Learning your own decisions teaches you nothing about anything else.
:dubious: Is that really a standard assumption? It wouldn’t be outlandish at all if our decisions correlated with outside events. Sometimes the fact that I choose to ask about so-and-so is strong evidence that such-and-such. You don’t need to resort to psychic powers to give a plausible psychological account of why you might be more likely to decide to ask certain questions in environments where the answer is “Yes”.
ETA: In other words, your “anti-psychic” assumption looks more to me like an assumption that our minds are somehow magically unaffected by the causal processes of the external material world.
Sure, there can be correlation, but not of a sort that increases the decision-maker’s own information about the external world.
It’s the following situation which is being ruled out:
John privately draws a card
John: Hey, Jill, what’s your credence for my having drawn the ace of clubs?
Jill: Well, right now, it’s 1/52.
John: Great. Now, sing a song to me.
Jill: Hm. Ok. sings Back in Black
Jill: Oh. I wonder why I picked the song Back in Black.
Jill: It must be my psychic affinity kicking in. Probably, your card is black.
Jill: I’m updating my credences. I now think there’s a 1/26 chance you drew the ace of clubs.
In my post #36, there is a line where I attribute to the reader the question “Did you eat cupcake #96?”. What I am saying linguistic convention rules out is for the actual reader, staring at those words on their computer, to proceed along the lines “Ah, well, since I am described as asking that question, perhaps this means there is something special about cupcake #96 which influenced me to ask about it; perhaps that was the only eaten cupcake, and I ended up asking about it not by marvelous coincidence, but by psychic affinity. Thus, whatever constructed probabilities I had for the cupcake-eating results prior to reading that question, I’ll now shift them accordingly, even though all I’ve learnt is the result of a private decision-making process I myself undertook.”
I’ll agree to this if you mean that that increase cannot happen prior to the decision’s being made.
Because after the decision is made, then I could look back on my own decision and notice that the fact that I decided the way I did implied something about the external world that I didn’t realize at the time. Not everything that causally influences my mind is known to me at the time of the influence.
Are we talking about the linguistic convention of probability puzzles or are we talking about the real world here? Because, sure, in the real world, it is possible that reflecting on decisions I made will tell me something I didn’t realize before about external facts; hell, it’s even possible that I am actually psychic. I have no quarrel with any of this.
My point is only that the linguistic convention of probability puzzles is to bar this out. That in the real world, such deduction may be possible is not in counter to this, because my point is specifically that this is a convention above and beyond what’s automatically thrust upon is.
. . . but I guess you just mean that my earlier decision was therefore not entirely rational, since I made it without conscious regard to information that was available to me at the time.
It’s not about rationality, as such, since there’s no need to bring in any framework of utility or goals. The assumption is just that any decision I make is probabilistically independent from all information I am unaware of.
Anyway, I think we’re getting stuck in an unimportant quagmire, since it was really a very minor observation being made in passing, rather than the thrust of anything.
By “rational”, in this case, I just mean that one is assigning credence in accordance with the conventional rules of probability.
But in regards to this formulation —
As far as your prior probability is concerned, this is true of any event in the story, not just your own decisions. It was the singling out of your own decisions for special treatment that I was objecting to. Your decisions as an agent within the story are events just like any other, and the probabilities that you assign to them should be as amenable to updating within the linguistic conventions of such probability puzzles.
Getting back to the cupcake problem . . .
So, it does make mathematical sense to condition on a proposition that simply describes an objective state of affairs, such as “I ate cupcake 96”. But, as others have mentioned, from an epistemological perspective, you should condition on everything that you know. In this case, you know not only that I ate cupcake 96; you also know that I told you that I ate cupcake 96. And so that is what an epistemic agent should condition on.
Many in this thread have pointed out that we can still get the >99% probability if we change the story so that the reader chooses which cupcake to ask about independently of the coin-flip.
This alternative version is a fine problem in its own right. Unfortunately for me, I have this ulterior motive having to do with the Sleeping Beauty paradox. Without getting into the SB paradox itself, the upshot is that I would like a reformulation of the cupcake paradox so that the propositions “cupcake 96 was eaten” and “I tell you that cupcake 96 was eaten” are logically equivalent (i.e., one is true if and only if the other is true).
(This isn’t the case in Indistinguishable’s version because, there, “I tell you that cupcake 96 was eaten” is equivalent to the conjunction of “cupcake 96 was eaten” AND “You ask about cupcake 96”.)
Sadly, the only ways that I can think of to make these propositions equivalent involve introducing SB-like weirdness (e.g., amnesia or cloned observers). It’s clear to me that the propositions become equivalent in these scenarios, but I know that this equivalence would be as controversial as the SB paradox itself.
Thanks to all for the replies so far. Can anyone now think of a nice uncontroversial way to make “cupcake 96 was eaten” and “I tell you that cupcake 96 was eaten” equivalent? That would just be the icing on my cupcake :).
Ok, fair enough. Everyone’s actions, by convention, are probabilistically independent of all information unknown to them. Was using “Your” rather than “Everyone’s” all that was bothering you about my original statement, or was there more to it?
ETA: Back to my not getting replies in at the appropriate times, I see…
You want the equivalence just for cupcake 96 or for all cupcakes? Either way, you can, of course, impose it by fiat, but I assume the conflict is that you don’t want the asymmetry of the former, but neither do you want the possibility in the latter of telling me about more than one cupcake being eaten.
Incidentally, I would be interested in seeing what your analogy between this and SB is. I assume the cupcakes represent days, and the ones which are eaten are the days on which Sleeping Beauty is awakened, but beyond that, I’m not sure what you’re trying to draw out.
The equivalence should be for all cupcakes. Actually, it would probably require such a significant change to the story that there’s no particular reason to be attached to cupcakes.
So, I guess I’m just looking for any plausible story wherein
(1) An epistemic agent learns that P because the agent is told that P, and
(2) P is logically equivalent to The agent was told that P.
(3) The agent knows (2) from the beginning of the story, prior to being told that P.
Any story like that? Surely that’s not all you’re after. One mechanically constructed story satisfying those properties is this:
John: I’m about to go into the next room, flip a coin, and then either eat all 100 cupcakes or just the blue one, accordingly. When I come back, I’ll tell you which cupcakes I ate.
Reader: Ok.
John: I’m back. Here are the numbers of the cupcakes of the cupcakes I ate: [blah blah blah].
We could take P to be “John eats cupcake X” and imagine this playing out in such a way as that John does eat cupcake X. For each cupcake, it is eaten just in case the Reader is told it is eaten (so condition (2) is satisfied). This whole setup was announced at the start (so condition (3) is satisfied). And the only way in which the Reader gains knowledge is through the announcements from John at the end (so condition (1) is satisfied).
But it’s too much; it’s trivial. At the end, the Reader is left with no uncertainty about anything (so far as the cupcake-eating goes), which presumably isn’t of any use for whatever analysis you’re trying to give. So I assume there are further constraints on what kind of story you want, only I don’t know what they are.
Yeah, you’re right, I don’t know how to give a concise list of minimal requirements. I’ll post again in a bit with brief explanation how I look at the SB paradox. That might make my needs clear.
You’re right about what the eaten cupcakes map to.
My response to the SB problem depends on whether, upon awakening, she has an objective means of designating the day of that awakening. If she doesn’t, then I’m a halfer. But if she does, then I’m a thirder.
For philosophical reasons, I hold that SB can’t designate the day of her awaking merely by saying “today” or “the day of this very utterance”. She at least needs to have (what I call) a “random calendar” for the duration of the experiment, or something equivalent.
A random calendar is a device with a display that shows one symbol per day in random order from a known finite set of symbols, never repeating a symbol. Hence, such a calendar has a finite span of days during which it is active. SB’s calendar must be known to be active throughout the longest-possible duration of the experiment. (It doesn’t matter, but, for simplicity, let’s suppose that her calendar is active for precisely that duration.)
With a random calendar, SB can now specify the current day by referring to it as “the day on which the calender displays <blah>”, where <blah> is the name of the symbol that she’s looking at. We assume that the first thing that SB does upon awakening is to look at the display on her random calendar. Only then does she determine her credence that the coin-flip came up heads.
Briefly, I’m a thirder in this case because SB’s credence that the coin came up heads should be the probability that the coin came up heads conditioned on the fact that the experiment’s duration included the day corresponding to the symbol on the calendar. This makes it more probable that the experiment lasted longer, and hence less probable that the coin came up heads.
On the other hand, if she doesn’t have a random calendar (or equivalent), she is not, on my view, able to condition on something like “the experiment’s duration included today”, because she has no way to make any particular day the referent of “today”. Hence, her credence that the coin came up heads remains the unconditioned probability of 1/2. (I can elaborate on why I’m a halfer in this case, but it’s not the case that relates to the cupcakes, so I’ll move on)
(Well, not before noting this: It follows from my view that, in the calendar-less case, she’s not able to assign a credence to the statement “It’s Monday” at all. If she uttered this statement, it would fail to refer to any particular day. Even though her utterance happens on a particular day, she has no way of specifying that day, so she can’t imbue her utterance with that meaning.)
Now for the connection to cupcakes. The underside labels on the cupcakes map to the symbols that the random calendar can display. The underside labels on the eaten cupcakes map to symbols that the random calendar displays during the experiment. (And the visible frosting-labels map to the actual dates of all these days.) Being told that cupcake 96 was eaten maps to waking up and seeing a particular symbol (the digits 96, say) on the calendar.
So, as I see it, SB is “told that cupcake 96 is eaten” if and only if it is in fact “eaten”, and she knows this. But I know that that view is controversial. Now, part of this controversy is because the story involves intuition-twisting things like amnesia. But part of the controversy is just because people are unhappy with the implication that SB assigns 1/3 credence to a fair coin-flip’s coming up heads. So I’m trying to cook up a story without things like amnesia in which something like this still happens.
Wow. That’s the first straight-up spambot post that I’ve ever seen here.
It won’t last more than 10 minutes. Reported.
Told ya 