- Imagine that we have a smart computer that accurately calculates the answers to probability questions that are put to it;
- a coin is flipped and the outcome recorded but not input into the computer data base;
- the computer has no clock and the memory is erased when it is turned off;
- when the computer is turned on it boots into exactly the same memory state;
- each time the computer is turned on it is given the information concerning this experiment;
- if the coin lands heads, then the computer is turned on once and asked to decide the "credence" that the coin is heads;
- if the coin lands tails, then the computer is turned on and asked to decide the "credence" that the coin is heads, it is then turned off and turned on again and asked to decide the "credence" that the coin is heads.
In other words, the anti-symmetric nature of the original problem is something of a red herring. The only analysis that the computer can give is that the probability that the coin was heads is 1/2 which is also the obvious answer.
Now let us consider a slightly different question: Which answer maximizes the expectation of being correct. We have the following slightly more complicated example:
- the probability of heads is p and tails is 1- p;
- if the coin lands heads, then we will interview the rebooted computer k times;
- if the coin lands tails, we will interview the rebooted computer n times;
- we will add 1 to the score for each correct guess by the computer;
- the computer will maximize the expected score.
- the computer answers heads at random with probability q and otherwise tails;
- if the coin lands heads, then the expected number of correct answers is qk;
- if the coin lands tails, then the expected number of correct answers is (1-q)n;
- the expected number of correct answers for the problem is pqk+(1-p)(1-q)n
In the case of the Sleeping Beauty Paradox, the expectation is easily seen to be 1 - (1/2)q. The expectation is maximized at 1 by always answering tails. This is, I believe, the source of the so-called paradox. The term credence is undefined and can be interpreted as expected correct answers or as the probability of being correct.
ReplyDeleteAs we have already seen the probability of being correct is always 1/2. But we expect to be correct 2 out of 3 interviews when we choose the answer that the coin landed tails.
If p=1/2, then the expectation is (1/2)(n+q(k-n)).
In this case, if k is less than n then the expectation is maximized by taking q=0, i.e. always answering tails.
If k equals n, then the strategy does not matter and the expectation is always n/2.
If k is greater than n, then the expectation is always maximized by answering heads.
I'm sorry, I thought you said the computer "accurately calculates the answers to probability questions." Because the answer to the first question is obviously 1/3. No, I'm not a newbie quack who hasn’t seen this problem before. I am, however, capable of setting aside any preconceived notions based on flawed intuition.
ReplyDeleteThe 1/2 answer was originated by David Lewis, a Professor of Philosophy (not Mathematics, as is obvious) from Princeton. His argument is essentially:
1) There are three events of importance, H1, T1, and T2. H vs. T indicates the result of the coin flip, and 1 vs. 2 indicates whether today is Monday or Tuesday, respectively.
2) Probability can be evaluated under two sets of circumstances (well, he had three, but he informed SB it was Monday after her initial assessment of probability): PS(*) on Sunday, before the experiment starts, and PX(*) during the experiment when she doesn’t know what day it is.
3) Since SB gained no useful information by being awakened - she knew it would happen - she can't change her assessment from PS(*) to PX(*). Hence PS(H1)=PX(H1)=1/2=PX(Heads).
This is absurd. First off, PS(H1)=PS(T1)=PS(T2)=0. PS is evaluated on Sunday, when it is impossible for "today" to be either Monday or Tuesday, as Lewis defined his events. Second, in order to use his logic structure (three significant events and two sets of circumstances to evaluate probability), he has to define his events so that they exist in both systems. If he had recognized this failing, he would have defined H1, T1, and T2 to be the events where the indicated coin result and the indicated day can be observed together. For SB on Sunday, these are PS(H1)=PS(T1)=PS(T2)=1/2.
"But wait a minute," I hear you cry. "Those probabilities add up to 1.5 for the three events!" Yes, they do. That's because Tails being observed on Monday (or your first computer reboot), which Lewis labeled T1, and Tails being observed on Tuesday (or the second reboot), which Lewis labeled T2, are the same event to an observer *on* *Sunday*. By the rules of the experiment, they are indistinguishable except to the person (or computer) inside the experiment. Lewis's assumption in Step 3, that SB gained no new useful information, is wrong. By being *inside* the experiment, T1 and T2 become disjoint events, allowing SB (or an accurate probability computer) to update the probability. P(Heads|Awake)=P(H1|Awake)/[P(H1|Awake)+P(T1|Awake)+ P(T2|Awake)]=(1/2)/(3/2)=1/3.
Thank you for your comments.
ReplyDeleteI have tried to avoid some of the complications of the
SB Paradox by restating the problem in a simplified form.
One of the problems inherent in the SBP is that imprecise
language is used to describe a very complex procedure.
I think that perhaps your statement that
"Lewis's assumption in Step 3, that SB gained no new useful information, is wrong. By being *inside* the experiment, T1 and T2 become disjoint events, allowing SB (or an accurate probability computer) to update the probability."
is an example that different people may make different and apparently equally plausible assumptions when the problem is ill-posed.
You may be right concerning your evaluation of the correct or proper assumption to be made but is a hard and fast rule of logic that any probability value must be less than one.
I would like to see your evaluation of the computer form of SBP and whether it matches your evaluation of the standard SBP.
"One of the problems inherent in the SBP is that imprecise language is used to describe a very complex procedure."
ReplyDeleteWhat's complex, or imprecise? You flip a coin, and you awaken SB once or twice based on the result. You arrange it so that during any awakening, she is unaware of any other possible awakenings. It's only "complex" if you feel the need to justify an answer that you can't justify without adding this complication you talk about.
"... is an example that different people may make different and apparently equally plausible assumptions when the problem is ill-posed."
The assumptions that lead to the answer 1/2 are implausible. Lewis himself assigned a non-zero probability to the statement, made on Sunday, that "It's Tuesday and the coin landed on Tails."
Try this simple variation: Everything is the same, except a second coin is flipped - the first one is a quarter, the second a nickel. If either coin lands on tails, the procedure is the same. If both land on heads, SB is left to sleep through Monday and awakened on Tuesday.
Any point in time during the experiment has an a priori chance of 1/4 to coincide with any one of the combinations {Heads&Mon, Heads&Tue, Tails&Mon, Tails&Tue}. But by being awake, SB knows that two of those are now be half as likely as the other two; further, that those two correspond to the quarter landing on heads. So P(Heads|Awake)=(1/8+1/8)/(1/8+1/8+1/4+1/4)=1/3.
Now, tell her the result of the nickel's flip. Are you claiming this answer changes? If you think so, please explain it to me, because requires some complication I just don’t see
As to the question of imprecision the original question is
ReplyDelete"What is your credence now for the proposition that the coin landed heads?"
Now does credence mean probability or does it mean expectation. These are different concepts entirely.
Those that argue on the basis of expectation will emphasize the three possible interviews while those that interpret this to mean probability will emphasize the fact that the coin flip has only two possible outcomes with equal probability
The paradox was inspired by the absent minded driver paradox see
AMD.
Most fields in mathematics can't handle ambiguity. You have to carefully define all of the parameter values that comprise any problem in the field. But one field not only handles it, it embraces ambiguity to the point that the entire field is based a measure of ambiguity. That measure is called *probability*, but it has no strict definition.
ReplyDeleteJust like the points and lines in geometry, it is described only by the set of properties we require it to have. These are called the Kolmogorov Axioms; simplified, they are: 1) Any set of outcomes (called an event) can be assigned a non-negative number called its probability, 2) The set of all possible outcomes has probability 1, and 3) the probability for the union of any two disjoint events is the sum of the probabilities of the individual events.
Any set of such numbers we assign to the set of all possible events is called a probability space. And there can be more than one such space - unconditional probability vs. conditional probability are two examples for the same process. But it is not the name we use - probability, credence, expectation, etc. - that distinguishes them. It is the relationships between the probabilities for the individual events, and how (or if) they make sense.
That relationship is not ambiguous in the Sleeping Beauty Problem, although some solvers try to use words like "credence," "centered probability," "Self-Indication," "Self-location," and "Self-sampling" as though they create such an ambiguity. But they don't; they are (unintentional, I hope) red herrings, which can be demonstrated by the fact that people who use the last three can't agree on what they mean, or how they affect the problem.
The 1/2 answer requires one of two assumptions that are incompatible with the SB problem as stated above. You could define the relationships between the events on Monday to be the same as the relationships on Sunday, but that is absurd because the set of events that are possible cannot be defined the same way in the two worlds. Lewis' concept of a "centered" event is required on Monday, but illogical on Sunday. Or you could define the SB on the two days to be different entities, so that not only are the memories separate, but the possible experiences are. This is sometimes known as "cloned SB" as opposed to "amnesia SB," since the two clones can't share the same experiences. But neither is consistent with your problem statement.
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With K=N, all your computer problem tells us, is that the computer does best by picking whichever result is more likely to be flipped. This is hardly surprising. It does not directly tell us the confidence the computer should place in Heads. But it does tell us the relative values. If N>K, as in the original problem, Q=0 maximizes your return. So your computer is telling us that it has more confidence in Tails than in Heads. This is inconsistent with saying its credence should be 1/2, and consistent with a credence for Tails that is greater than the credence for Heads.
I haven't studied it closely; but if the coin is biased at P=N/(N+K), it doesn't matter what Q is. Your expected value is N/(N+K) regardless of Q. This probably means that the confidence in Heads should be K/(N+K), which is the thirder solution.
Thanx for your comments. I find them very constructive and I will
ReplyDeleteread them with care and try to incorporate your ideas in the future.