I read it as “What is your credence”, which is supposed to be synonymous with “subjective probability”, which—and this is significant—I take to entail that Beauty must condition on having been woken (because she conditions on every piece of information known to her).
In other words, I take the question to be precisely “What is the probability you assign to the coin having come up heads, taking into account your uncertainty as to what day it is.”
Ahhhh, I think I understand a bit better now. Am I right in thinking that your objection is not that you disapprove of relative frequency arguments in themselves, but that you believe the wrong relative frequency/frequencies is/are being used?
Right up until your reply prompted me to write a program to check your argument, I wasn’t thinking in terms of relative frequencies at all, but in terms of probability distributions.
I haven’t learned the rules for relative frequencies yet (by which I mean thing like “(don’t) include counts of variables that have a correlation of 1 in your denominator”), so I really have no idea.
Here is my program—which by the way agrees with neq1′s comment here, insofar as the “magic trick” which will recover 1⁄2 as the answer consists of commenting out the TTW line.
However, this seems perfectly nonsensical when transposed to my spreadsheet: zeroing out the TTW cell at all means I end up with a total probability mass less than 1. So, I can’t accept at the moment that neq1′s suggestion accords with the laws of probability—I’d need to learn what changes to make to my table and why I should make them.
from random import shuffle, randint
flips=1000
HEADS=0
TAILS=1
# individual cells
HMW = HTW = HMD = HTD = 0.0
TMW = TTW = TMD = TTD = 0.0
def run_experiment():
global HMW, HTW, HMD, HTD, TMW, TTW, TMD, TTD
coin = randint(HEADS,TAILS)
if (coin == HEADS):
# wake Beauty on monday
HMW+=1
# drug Beauty on Tuesday
HTD+=1
if (coin == TAILS):
# wake Beauty on monday
TMW+=1
# wake Beauty on Tuesday too
TTW+=1
for i in range(flips):
run_experiment()
print "Total samples where heads divided by total samples ~P(H):",(HMW+HTW+HMD+HTD)/(HMW+HTW+HMD+HTD+TMW+TTW+TMD+TTD)
print "Total samples where woken F(W):",HMW+HTW+TMW+TTW
print "Total samples where woken and heads F(W&H):", HMW+HTW
print "P(W&H)=P(W)P(H|W), so P(H|W)=lim F(W&H)/F(W)"
print "Total samples where woken and heads divided by sample where woken F(H|W):", (HMW+HTW)/(HMW+HTW+TMW+TTW)
Replying again since I’ve now looked at the spreadsheet.
Using my intuition (which says the answer is 1⁄2), I would expect P(Heads, Tuesday, Not woken) + P(Tails, Tuesday, Not woken) > 0, since I know it’s possible for Beauty to not be woken on Tuesday. But the ‘halfer “variant”’ sheet says P(H, T, N) + P(T, T, N) = 0 + 0 = 0, so that sheet’s way of getting 1⁄2 must differ from how my intuition works.
(ETA—Unless I’m misunderstanding the spreadsheet, which is always possible.)
Your program looks good here, your code looks a lot like mine, and I ran it and got ~1/2 for P(H) and ~1/3 for F(H|W). I’ll try and compare to your spreadsheet.
I read it as “What is your credence”, which is supposed to be synonymous with “subjective probability”, which—and this is significant—I take to entail that Beauty must condition on having been woken (because she conditions on every piece of information known to her).
In other words, I take the question to be precisely “What is the probability you assign to the coin having come up heads, taking into account your uncertainty as to what day it is.”
Ahhhh, I think I understand a bit better now. Am I right in thinking that your objection is not that you disapprove of relative frequency arguments in themselves, but that you believe the wrong relative frequency/frequencies is/are being used?
Right up until your reply prompted me to write a program to check your argument, I wasn’t thinking in terms of relative frequencies at all, but in terms of probability distributions.
I haven’t learned the rules for relative frequencies yet (by which I mean thing like “(don’t) include counts of variables that have a correlation of 1 in your denominator”), so I really have no idea.
Here is my program—which by the way agrees with neq1′s comment here, insofar as the “magic trick” which will recover 1⁄2 as the answer consists of commenting out the TTW line.
However, this seems perfectly nonsensical when transposed to my spreadsheet: zeroing out the TTW cell at all means I end up with a total probability mass less than 1. So, I can’t accept at the moment that neq1′s suggestion accords with the laws of probability—I’d need to learn what changes to make to my table and why I should make them.
Replying again since I’ve now looked at the spreadsheet.
Using my intuition (which says the answer is 1⁄2), I would expect P(Heads, Tuesday, Not woken) + P(Tails, Tuesday, Not woken) > 0, since I know it’s possible for Beauty to not be woken on Tuesday. But the ‘halfer “variant”’ sheet says P(H, T, N) + P(T, T, N) = 0 + 0 = 0, so that sheet’s way of getting 1⁄2 must differ from how my intuition works.
(ETA—Unless I’m misunderstanding the spreadsheet, which is always possible.)
Yeah, that “Halfer variant” was my best attempt at making sense of the 1⁄2 answer, but it’s not very convincing even to me anymore.
That program is simple enough that you can easily compute expectations of your 8 counts analytically.
Your program looks good here, your code looks a lot like mine, and I ran it and got ~1/2 for P(H) and ~1/3 for F(H|W). I’ll try and compare to your spreadsheet.