文章探讨了睡美人问题，认为正确答案是Halfer立场（硬币正面朝上的概率为1/2），并从多个角度批判了Thirder立场，包括贝叶斯推理、频率主义批判、自我定位信念、人类中心推理等方面，强调了概率计算的正确原则。

🧐文章指出睡美人问题的正确答案是Halfer立场，认为Thirder立场基于对贝叶斯和频率主义推理的错误应用。在贝叶斯推理中，醒来事件在两种情况下同样可能，不能提供更新先验的新信息，所以原有的50/50概率在醒来后应保持不变。

📚在频率主义批判方面，文章认为Thirder立场常错误地应用频率主义概率。在睡美人问题中，主观上醒来的体验无论实际发生次数如何都是相同的，将主观上无法区分的事件视为不同来计算概率是根本错误的。

💡文章还探讨了自我定位信念、人类中心推理等方面对Thirder立场的批判。认为自我定位信息与硬币翻转概率无关，人类中心推理将不同事件的结果错误地结合，这些都是错误的。

🎲此外，文章讨论了最优投注策略与概率的关系。指出虽然在某些情况下总是投注Tails是最优策略，但这并不意味着Tails的概率更高，而是因为回报结构的不对称，硬币翻转的基础概率仍然是50/50。

Published on October 16, 2024 10:20 AM GMT

Preamble

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

Motivation

I was recently introduced to the canonical Sleeping Beauty problem and initially was a halfer but confused. Or more like I thought the halfer position was correct, but smart people seemed to be thirders and I was worried I was misunderstanding something about the problem, or confused myself or similar.

I debated the problem extensively on the LW Discord server and with some LLMs and strongly updated towards "thirders are just engaging in gross epistemic malpractice".

A message I sent in the LW server:

Alternatively I started out confused.

Debating this problem here and with LLMs convinced me that I'm not confused and the thirders are actually just doing epistemological nonsense.

It feels arrogant, but it's not a poor reflection of my epistemic state?

I still have some meta level uncertainty re: the nonsense allegations.

I want to be convinced that the thirder position is not nonsense and there is a legitimate disagreement/debate to be had here.

I have read some of the LW posts on the canonical problem here. I won't be linking them due to laziness.

I asked Sonnet 3.5 to distill my position/rejections from our debate and below is its summary.

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Comprehensive Position on the Sleeping Beauty Problem

1. Core Position

The correct answer to the Sleeping Beauty problem is the Halfer position (1/2 probability for Heads).The Thirder position (1/3 probability for Heads) is based on misapplications of both Bayesian and frequentist reasoning.

2. Bayesian Reasoning

Probability is a property of the map (agent's beliefs), not the territory (environment).For an observation O to be evidence for a hypothesis H, P(O|H) must be > P(O|¬H).The wake-up event is equally likely under both Heads and Tails scenarios, thus provides no new information to update priors.The original 50/50 probability should remain unchanged after waking up.

3. Frequentist Critique

The Thirder position often relies on a misapplication of frequentist probability.Multiple wake-ups in the Tails world are irrelevant if they're subjectively indistinguishable from a single wake-up.Counting indistinguishable events as separate for probability calculations is fundamentally flawed.

Expanded Argument:

Frequentist probability is based on the long-run frequency of events. However, in the Sleeping Beauty problem, the subjective experience of waking up is identical whether it happens once or multiple times.Consider two scenarios:
Sleeping Beauty is woken once in the Heads world and once in the Tails world.Sleeping Beauty is woken once in the Heads world and a million times in the Tails world.
From Beauty's perspective, these scenarios are indistinguishable. She experiences one wake-up event, regardless of how many times it actually occurs.Thirders often argue that in scenario 2, the probability of Tails approaches 1 as the number of Tails wake-ups increases. This is a misapplication of frequentist reasoning because:a) It treats subjectively indistinguishable events as distinct.b) It conflates the frequency of wake-ups with the probability of the coin flip outcome.The correct application of frequentist probability should focus on the coin flip itself, which remains a 50/50 proposition regardless of the wake-up protocol.

Counterargument to Thirder Position:

Thirders might argue: "If we repeated this experiment many times, Beauty would find herself in a Tails wake-up twice as often as a Heads wake-up."Rebuttal: This argument incorrectly treats each wake-up as a separate trial of the experiment. In reality, each complete experiment (coin flip + wake-ups) is one trial, and in each trial, the coin has a 50% chance of landing Heads.

4. Self-Locating Beliefs

Self-locating information (which wake-up you're experiencing) is irrelevant to the coin flip probability.The question "What is the probability of Heads?" is about the coin, not about your location in time or possible worlds.

5. Anthropic Reasoning Rejection

Anthropic arguments that treat all possible wake-ups as equally likely samples are rejected.This approach incorrectly combines outcomes from distinct events (coin flip and wake-up protocol).

Expanded Argument:

Anthropic reasoning in this context suggests that Beauty should consider herself as randomly selected from all possible wake-up events.This reasoning is flawed because:
It treats the wake-up events as the primary random process, when the actual random process is the coin flip.It conflates the sampling process (how Beauty is woken up) with the event we're trying to determine the probability of (the coin flip).

Specific Anthropic Argument and Counterargument:

Anthropic Argument: "When Beauty wakes up, she is essentially sampling from the space of all possible wake-ups. There are twice as many Tails wake-ups as Heads wake-ups, so the probability of Heads is 1/3."Counterargument:
This incorrectly assumes that each wake-up is an independent event, when they are actually dependent on a single coin flip.It ignores the fact that the probability we're interested in is that of the coin flip, not the wake-up event.This reasoning would lead to absurd conclusions if we changed the wake-up protocol (e.g., waking Beauty a million times for Tails would make Heads virtually impossible, which is clearly wrong).

6. Distinguishability vs. Probability

Subjective indistinguishability of events doesn't imply equal probability of the underlying states.However, indistinguishability means the events can't provide evidence for updating probabilities.

7. Betting Strategies vs. Probabilities

Optimal betting strategies (e.g., always bet on Tails) don't necessarily reflect true probabilities.Asymmetric payoffs can justify betting on Tails without changing the underlying 50/50 probability.

Expanded Argument:

The Sleeping Beauty problem presents a scenario where the optimal betting strategy (always betting on Tails) seems to contradict the claimed 50/50 probability. This apparent contradiction is resolved by recognizing that:
Betting strategies can be influenced by factors other than pure probability, such as payoff structures.The expected value of a bet is not solely determined by the probability of an event, but also by the payoff for each outcome.In this case, the Tails outcome provides more opportunities to bet, creating an asymmetry in the payoff structure.

Specific Example:

Consider a simplified version of the problem where:
If the coin lands Heads, Beauty is woken once and can bet $1.If the coin lands Tails, Beauty is woken twice and can bet $1 each time.The payoff for a correct bet is 1:2 (you double your money).
The optimal strategy is to always bet on Tails, because:
Betting on Heads: 50% chance of winning $1 = $0.50 expected valueBetting on Tails: 50% chance of winning $2 (betting twice) = $1.00 expected value
However, this doesn't mean the probability of Tails is higher. It's still 50%, but the payoff structure makes betting on Tails more profitable.

Analogy to Clarify:

Imagine a fair coin flip where you're offered the following bet:
If you bet on Heads and win, you get $1.If you bet on Tails and win, you get $K (where K >> 1, i.e., K is much larger than 1).
The optimal strategy is to bet on Tails every time, even though the coin is fair (50/50).If you repeat this experiment many times, always betting on Tails will be a winning strategy in the long run.Despite this, the probability of the coin landing Heads remains 0.5 (50%).

Counterargument to Thirder Position:

Thirders might argue: "The optimal betting strategy aligns with the 1/3 probability for Heads."Rebuttal: This confuses expected value with probability. The betting strategy is optimal due to the asymmetric nature of the payoffs (betting twice on Tails vs. once on Heads), not because Tails is more likely. The underlying probability of the coin flip remains 50/50, regardless of the betting structure.

8. Counterfactuals and Different Problems

Arguments involving additional information change the problem fundamentally."X & Y is evidence for H, therefore X is evidence for H" is invalid reasoning.

9. Information Relevance

Not all information about the experimental setup is relevant for probability calculations.The wake-up protocol, while part of the setup, doesn't provide discriminatory evidence for Heads vs. Tails.

10. Epistemological Stance

Adheres to strict Bayesian principles for updating beliefs.Rejects arguments that conflate distinct problems or misapply probabilistic concepts.

11. Common Thirder Arguments Addressed

Frequency of wake-ups: Irrelevant due to subjective indistinguishability.Anthropic reasoning: Incorrectly combines distinct events.Betting strategies: Don't necessarily reflect true probabilities.Self-locating beliefs: Irrelevant to the coin flip probability.

12. Meta-level Considerations

Many arguments for the Thirder position stem from subtle misapplications of otherwise valid probabilistic principles.

13. Openness to Counter-Arguments

Willing to consider counter-arguments that adhere to rigorous Bayesian principles.Rejects arguments based on frequentist interpretations, anthropic reasoning, or conflation of distinct problems.

This position maintains that the Sleeping Beauty problem, when correctly analyzed using Bayesian principles, does not provide any new information that would justify updating the prior 50/50 probability of the coin flip. It challenges readers to present counter-arguments that do not rely on commonly rejected reasoning patterns and that strictly adhere to Bayesian updating based on genuinely new, discriminatory evidence.

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