I appreciate your input nonetheless! I love these sorts of things, even though I often need help getting across the finish line. (The ace of spades example in the Monty Hall problem was my son’s idea. It’s a really helpful way of thinking about it.)
What if we modify the Sleeping Beauty problem thusly: on tails she will be woken twice and intervened as before, but for heads she will be woken twice, but only one of those times she’ll be interviewed?
For the 100 women problem: what if the causality is flipped? That is, no women are brought in until the coin is flipped. Heads, one is sought out, instructed and interviewed; tails, 100 are sought out and interviewed?
Does the shift from “credence” to “probability” shift the question from one about the coin toss itself to one of a likelihood within the system as a whole?
As far as I can tell, your first variation seems similar to my second one, but there seems to be a catch. In your version, SB knows that for a tails outcome, she will be interviewed for every waking experience, but for a heads outcome she will be interviewed only half the time. This means that, in any non-interview scenario, SB will know in that moment that the coin came up heads. She also knows, in any interview scenario, that she is not in the non-interview scenario.
However, unlike my second 100 volunteers scenario, SB will definitely be interviewed at least once regardless of the outcome of the coin toss—and this, as far as I can tell, means that knowledge that she is not in the non-interview scenario conveys no information. Even if you make it 100 wakings with 1 interview vs. 100 wakings with 100 interviews, the same individual will inevitably be interviewed no matter how the coin comes up.
As far as I can see, then, when SB is interviewed, she can only fall back on the mathematical default: The likelihood of a fair coin coming up heads is 1/2, and that’s all she knows. However, if you ask her to *guess* the outcome of the coin toss, and her goal is to maximize correct guesses and minimize wrong ones, then of course she should guess the outcome that results in the most interviews.
Your second variation seems to me even more straightforward. If you toss a coin and then recruit a volunteer what they think is the outcome, they have no reason to lean either way. And the same is true if you recruit 100 volunteers, assuming of course that you *either* don’t tell them which outcome was connected with recruiting 100 volunteers *or* don’t tell them how many volunteers you have recruited!
I *think* your approach makes sense, but haven't had time to go through the problem in detail. I agree that the central problem here is being exacerbated by vagueness around the meaning of "credence". I was immediately doubtful about that word.
My inclination whenever I see a problem like Monty Haul is to check if someone has coded the problem. It would be pretty easy to write a couple dozen lines of python to do a few thousand simulation runs and accumulate the probability results. Seeing someone do that is actually what finally made the Monty Haul problem "click" for me.
I look forward to your considered thoughts when you’ve had a chance to wrap your head around the issues! Thanks for your thoughtful reading and comments.
Was starting to read opinions online, and in this comments section, but just realized that might bias me, so I stopped. Took a fresh description of the problem and here's the thinking I did:
-How likely was it that the coin was heads (or tails)? 1/2
-There are 3 ways that SB can have been woken up. 1 from heads and 2 from tails.
-How likely was it that SB is being interviewed with a heads-based wake up? 1/3
-How likely was it that SB is being interviewed with a tails-based Monday wake up? 1/3
-How likely was it that SB is being interviewed with a tails-based Tuesday wake up? 1/3
-How likely was it that SB is being interviewed with any tails-based wake up? 1/3 + 1/3 = 2/3
That feels... plausible, I think. But I'm still very uncomfortable with what the word "credence" means...
Let's try a variation, but using the same logic:
Variation 1
On heads, SB is woken up once. On tails, SB is woken up 99 times. What is her credence that it landed heads?
-There are 100 ways that SB can have been woken up. 1 from heads and 99 from tails.
-How likely was it that the coin was heads (or tails)? 1/2
-How likely was it that SB is being interviewed with a heads-based wake up? 1/100
-How likely was it that SB is being interviewed with any specific tails-based wake up? 1/100
-How likely was it that SB is being interviewed with any tails-based wake up? 99/100
So SB would say the coin had been heads 1% of the time. This feels... wrong. Those 99 wake ups feel extraneous to the problem.
Variation 2
-Original problem (1 wake up or 2), but it is run 100 times in a row on poor SB, and we get 50 heads and 50 tails. In total, SB has been woken up 150 times. In 50 of those interviews, the coin had actually been heads, and in 100 of them, it had been tails. Again, SB saying that she feels like the coin was 1/3 heads seems plausible.
Variation 3
-Combine them. 1 wake up on heads, 99 wake ups on tails. Run a hundred times in a row, with 50 heads and 50 tails. In total, SB is woken 50+(50*99)=5000 times. In 50 of those interviews, the coin had been heads, and 4950 of them, it had been tails. SB saying the coin was heads 1% of the time now feels right to me again. (while still feeling kind of wrong)
I try to think about this problem as little as possible because it makes my head hurt, but my tentative view is that the question itself is ambiguous. I'd argue it's unclear whether it's asking for P(heads) — the probability that the coin landed heads — or P(heads | awake) — the probability that the coin landed heads given that she is awake. The former is 1/2; the latter is 1/3.
Counterpoint: As I see it, the question is not P(heads | awake) or P(tails| awake), but P(heads | awake) vs. P(tails| awake1) vs. P(tails| awake2). And it seems to me that the odds of each of these three scenarios is P(heads | awake) = 1/2, P(tails| awake1) = 1/4, and P(tails| awake2) = 1/4. Therefore, SB should answer that her credence for the proposition that the head landed heads is 1/2!
I rear back in terror at the sight of these sorts of things, so: I can't help you!
I appreciate your input nonetheless! I love these sorts of things, even though I often need help getting across the finish line. (The ace of spades example in the Monty Hall problem was my son’s idea. It’s a really helpful way of thinking about it.)
So many questions.
What if we modify the Sleeping Beauty problem thusly: on tails she will be woken twice and intervened as before, but for heads she will be woken twice, but only one of those times she’ll be interviewed?
For the 100 women problem: what if the causality is flipped? That is, no women are brought in until the coin is flipped. Heads, one is sought out, instructed and interviewed; tails, 100 are sought out and interviewed?
Does the shift from “credence” to “probability” shift the question from one about the coin toss itself to one of a likelihood within the system as a whole?
Thanks for the new variations, M L!
As far as I can tell, your first variation seems similar to my second one, but there seems to be a catch. In your version, SB knows that for a tails outcome, she will be interviewed for every waking experience, but for a heads outcome she will be interviewed only half the time. This means that, in any non-interview scenario, SB will know in that moment that the coin came up heads. She also knows, in any interview scenario, that she is not in the non-interview scenario.
However, unlike my second 100 volunteers scenario, SB will definitely be interviewed at least once regardless of the outcome of the coin toss—and this, as far as I can tell, means that knowledge that she is not in the non-interview scenario conveys no information. Even if you make it 100 wakings with 1 interview vs. 100 wakings with 100 interviews, the same individual will inevitably be interviewed no matter how the coin comes up.
As far as I can see, then, when SB is interviewed, she can only fall back on the mathematical default: The likelihood of a fair coin coming up heads is 1/2, and that’s all she knows. However, if you ask her to *guess* the outcome of the coin toss, and her goal is to maximize correct guesses and minimize wrong ones, then of course she should guess the outcome that results in the most interviews.
Your second variation seems to me even more straightforward. If you toss a coin and then recruit a volunteer what they think is the outcome, they have no reason to lean either way. And the same is true if you recruit 100 volunteers, assuming of course that you *either* don’t tell them which outcome was connected with recruiting 100 volunteers *or* don’t tell them how many volunteers you have recruited!
My thoughts:
#1 Let's Make a Deal — Goats are cute, so I'll be happy either way.
#2 Boy/Girl Puppy — I don't have the energy to housebreak a puppy right now, so the point is moot.
#3 Sleeping Beauty — Women are sleep-deprived enough. Let the woman sleep from Sunday through Wednesday.
I love it! These are great responses. Thanks so much, Karen!
"If it comes up tails, we will ask ask all 100 of them what their credence is that the coin came up heads, and the other 99 will be asked nothing."
Am I failing to parse this? There aren't 199 people here, right?
Editing glitch: Looks like that last clause belonged with the previous sentence. Does it make sense now?
Yep, that works.
So you think my approach makes sense?
I *think* your approach makes sense, but haven't had time to go through the problem in detail. I agree that the central problem here is being exacerbated by vagueness around the meaning of "credence". I was immediately doubtful about that word.
My inclination whenever I see a problem like Monty Haul is to check if someone has coded the problem. It would be pretty easy to write a couple dozen lines of python to do a few thousand simulation runs and accumulate the probability results. Seeing someone do that is actually what finally made the Monty Haul problem "click" for me.
I look forward to your considered thoughts when you’ve had a chance to wrap your head around the issues! Thanks for your thoughtful reading and comments.
Was starting to read opinions online, and in this comments section, but just realized that might bias me, so I stopped. Took a fresh description of the problem and here's the thinking I did:
-How likely was it that the coin was heads (or tails)? 1/2
-There are 3 ways that SB can have been woken up. 1 from heads and 2 from tails.
-How likely was it that SB is being interviewed with a heads-based wake up? 1/3
-How likely was it that SB is being interviewed with a tails-based Monday wake up? 1/3
-How likely was it that SB is being interviewed with a tails-based Tuesday wake up? 1/3
-How likely was it that SB is being interviewed with any tails-based wake up? 1/3 + 1/3 = 2/3
That feels... plausible, I think. But I'm still very uncomfortable with what the word "credence" means...
Let's try a variation, but using the same logic:
Variation 1
On heads, SB is woken up once. On tails, SB is woken up 99 times. What is her credence that it landed heads?
-There are 100 ways that SB can have been woken up. 1 from heads and 99 from tails.
-How likely was it that the coin was heads (or tails)? 1/2
-How likely was it that SB is being interviewed with a heads-based wake up? 1/100
-How likely was it that SB is being interviewed with any specific tails-based wake up? 1/100
-How likely was it that SB is being interviewed with any tails-based wake up? 99/100
So SB would say the coin had been heads 1% of the time. This feels... wrong. Those 99 wake ups feel extraneous to the problem.
Variation 2
-Original problem (1 wake up or 2), but it is run 100 times in a row on poor SB, and we get 50 heads and 50 tails. In total, SB has been woken up 150 times. In 50 of those interviews, the coin had actually been heads, and in 100 of them, it had been tails. Again, SB saying that she feels like the coin was 1/3 heads seems plausible.
Variation 3
-Combine them. 1 wake up on heads, 99 wake ups on tails. Run a hundred times in a row, with 50 heads and 50 tails. In total, SB is woken 50+(50*99)=5000 times. In 50 of those interviews, the coin had been heads, and 4950 of them, it had been tails. SB saying the coin was heads 1% of the time now feels right to me again. (while still feeling kind of wrong)
Thoughts?
I try to think about this problem as little as possible because it makes my head hurt, but my tentative view is that the question itself is ambiguous. I'd argue it's unclear whether it's asking for P(heads) — the probability that the coin landed heads — or P(heads | awake) — the probability that the coin landed heads given that she is awake. The former is 1/2; the latter is 1/3.
Thanks, Dale!
Counterpoint: As I see it, the question is not P(heads | awake) or P(tails| awake), but P(heads | awake) vs. P(tails| awake1) vs. P(tails| awake2). And it seems to me that the odds of each of these three scenarios is P(heads | awake) = 1/2, P(tails| awake1) = 1/4, and P(tails| awake2) = 1/4. Therefore, SB should answer that her credence for the proposition that the head landed heads is 1/2!
Counterpoint to your counterpoint: I hate this problem.
LOL! FTR, I’m not sure I’m right! But the more I think about it, the less any other approach makes sense to me. I’m open to being convinced otherwise!