Lesson · Paradoxes and Their Exits · Probability

Sleeping Beauty

Beauty is put to sleep on Sunday and a fair coin is tossed. On heads she is woken once, on tails twice, with memory of the first waking erased. Woken, she is asked how likely heads is. The halfer says one half, the thirder one third. Each is a correct update from a stated likelihood, and what the lab shows is that they differ in the likelihood and nowhere else.

Open the interactive lesson Practice

heads: wake Monday. tails: Monday, Tuesday posterior ∝ prior × likelihood halfer: P(awake | H) = P(awake | T) = 1 thirder: P(this waking | H) = 1/2 same rule, two likelihoods

Two answers, one rule

Core ideas

What this lesson establishes

The puzzle is not about arithmetic. Three ideas show what it is about.

The prior is not in dispute

On Sunday the coin is fair, so the credence in heads is one half. Everyone agrees. The question is whether being woken is evidence that should move it.

The evidence is stated by a likelihood

Bayes' rule needs the probability of the evidence under each hypothesis. “I am awake” can be modelled as certain on both outcomes of the coin, or as a particular waking whose probability differs between them. Those are the two models, and the rule is the same for both.

The disagreement is about what the evidence is

The halfer treats being awake as something she knew in advance would happen. The thirder treats it as locating her at one of the possible wakings, which tails produces twice as often. Both are positions about a self-locating fact, and no arithmetic chooses between them.

The material

One rule and two likelihoods

The setup, the two answers, and the lab that shows what they share.

On Sunday evening Beauty is told the plan, and she is put to sleep. A fair coin is tossed. If it lands heads she is woken on Monday and the experiment ends. If it lands tails she is woken on Monday, put back to sleep with a drug that erases the memory of the waking, and woken again on Tuesday. Each time she is woken she is asked: how likely is it that the coin landed heads? She cannot tell Monday's waking from Tuesday's, and she knows all this in advance.

The halfer answers one half. She knew on Sunday that she would be woken, and being woken tells her nothing she did not know, so her credence in heads should be what it was. The thirder answers one third. Over many repetitions of the experiment a third of her wakings follow heads, and since she cannot tell which waking this is, one third is the credence that matches what her wakings are like.

Both answers can be written as an update by Bayes' rule, as in “Updating on Evidence”. There are two hypotheses, heads and tails, each with prior one half. What differs is the likelihood of the evidence. The halfer says the evidence is “I am awake”, which has probability 1 if the coin is heads and probability 1 if it is tails, since she is woken either way.

P(awake | H) = 1 P(awake | T) = 1 Bayes factor = 1/1 = 1 P(H | awake) = 1/2

The thirder says the evidence is the fact that this waking is occurring. Consider a day on which she might be woken. Under heads only one of the two days, Monday, has a waking, so the chance that this day is a waking day is one half. Under tails both days have one, so it is 1. The likelihoods are one half and 1.

P(this waking | H) = 1/2 P(this waking | T) = 1 Bayes factor = (1/2)/1 = 1/2 P(H | this waking) = 1/3

The lab shows the two computations. In the first preset, the halfer's, the Bayes factor is 1 and the posterior for heads stays at one half. In the second, the thirder's, the Bayes factor is one half, the evidence disconfirms heads, and the posterior is one third. The rule that produced both is the same, and so is the prior. Only the likelihood of the word “awake” was changed.

Example · Where the two models part

Ask what each says when Beauty learns that it is Monday. The thirder starts from one third for heads and the likelihood of Monday is 1 under heads and one half under tails, which brings her to one half. The halfer starts from one half with the same likelihoods and goes to two thirds. The third preset is the thirder's, and editing its prior to one half and one half gives the halfer's. A fair coin that has not yet been tossed, in the version where it is tossed on Monday night, moves to two thirds on the news that it is Monday, and that is what the halfer has to defend.

Each position can be put at its strongest. The halfer has a clean principle behind her: evidence is something you did not already know, and Beauty knew she would be woken. Her cost is the case just given, where a fair coin she has not seen tossed moves to two thirds on the news of a weekday. The thirder has a clean principle too: credences should be right on average over the occasions on which they are held. Her cost is a credence of one third in a coin she knows is fair, reached by evidence that is not about the coin in the ordinary sense.

So the exit is not the usual one. Set out as this course sets out a paradox, there are two arguments, each valid, each from the Sunday prior and a likelihood, to credences that cannot both be held; the exit is to deny a premise of one of them, and the only premise that differs is the likelihood. Neither side denies a premise of Bayes' rule or faults a step of it. The choice is of a likelihood model, and each model implies the whole of its answer. This is why the question cannot be settled by checking the sum, and why the lesson does not announce a winner. Whether a self-locating fact, “this is a waking”, counts as evidence about the coin is the premise the two answers divide on.

The limit of the lab is that it computes from a likelihood and does not decide which to put in. A bet does not settle it either unless it says what is scored. If a stake is paid for each waking, tails pays twice as often as heads and betting on tails is the better bet whichever model is held. If one stake is paid for the experiment, the bet is even. That the best bet depends on the scoring, and not on the model alone, is a further fact about the puzzle and not a way round it.

Interactive

Exact Bayesian updating

prior times likelihood, normalised; the Bayes factor is the weight of the data

The same rule with two likelihoods

The hypotheses are heads and tails, and the tiles report heads. In the first two presets the prior is the Sunday one, one half each, and only the likelihood of the evidence changes. In the last two Beauty is also told that it is Monday, which has likelihood 1 under heads and one half under tails; the prior is the thirder's one third in the first and the halfer's one half in the second.

Worked example

Prior, one per hypothesis

Likelihoods: a row per hypothesis, a column per outcome, rows separated by “;”

Data: outcome names separated by spaces (red*10 is ten reds)

Payoffs, “act: one per hypothesis; …” (empty for none)

The hypothesis the tiles report

Posterior—

Bayes factor—

The data—

Next outcome is the first—

Best act now—

Method

Comparing two answers to a puzzle of evidence

Four steps, for Sleeping Beauty and for any puzzle where the answers differ and the rule is the same.

Fix the prior

State the hypotheses and the credence in each before the evidence. Here heads and tails at one half each, which both sides accept.

Write each side's evidence as a likelihood

For each hypothesis, give the probability of the evidence as that side describes it. The two sides will differ here, if anywhere.

Compute both posteriors by the same rule

Run the update for each. If both are correct computations, the disagreement is already located in the likelihoods.

State what the disagreement is about

Say which likelihood each side defends and what it costs. A test case that separates them, such as being told it is Monday, helps.

Likelihoods and answers

Score 0 / 0

Choose the answer you can justify.

Heads, under two likelihoods

The prior is one half for heads. The question is what the evidence “I am awake” does to it under each model.

prior: heads 1/2, tails 1/2 halfer: P(awake | heads) = 1, P(awake | tails) = 1 Bayes factor = 1/1 = 1 posterior for heads = 1/2 thirder: P(this waking | heads) = 1/2, | tails = 1 Bayes factor = (1/2)/1 = 1/2 posterior for heads = 1/3 same rule, same prior, different likelihood

The halfer's Bayes factor is 1 and her posterior one half. The thirder's Bayes factor is one half and her posterior one third. Neither made a mistake in the arithmetic.

Common mistakes

Thinking the coin is fair, so the answer must be one halfFairness fixes the prior, and nobody disputes it. The question is what being woken does to the prior, and that depends on the likelihood of the evidence under each outcome. With a likelihood of 1 for both, the posterior is one half. With one half under heads and 1 under tails, it is one third, for the same fair coin.

Taking the two answers to be an error of arithmetic on one sideEach answer is a correct update from its likelihood, and the lab shows both. The dispute is over the likelihood, which is a statement of what the evidence is, and checking the sum cannot settle it.

Expecting a bet to settle itA bet is scored in some way, per waking or per experiment, and the best bet follows from the scoring. Per waking, tails wins twice as often. Per experiment, the bet is even. Which scoring matches Beauty's credence is part of the question.

Check your understanding

✓

Finish when you can compute both answers by one rule and say what the disagreement is about.

Given the setup, you should be able to write the likelihood each side uses, compute the Bayes factor and posterior for heads under each, run a test case such as being told it is Monday, and state that the dispute is over the likelihood of the evidence.

Note: This lesson makes no claim about which answer is right. Real work on the puzzle weighs the two costs named above, and some reject both models in favour of a third. The next lesson returns to a puzzle where the answer is settled once the model is stated, and the interest is in how a small change in the model moves it.