Beauty quips, "I'd shut up and multiply!"

6 neq1 07 May 2010 02:34PM

When it comes to probability, you should trust probability laws over your intuition.  Many people got the Monty Hall problem wrong because their intuition was bad.  You can get the solution to that problem using probability laws that you learned in Stats 101 -- it's not a hard problem.  Similarly, there has been a lot of debate about the Sleeping Beauty problem.  Again, though, that's because people are starting with their intuition instead of letting probability laws lead them to understanding.

The Sleeping Beauty Problem

On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.

Each interview consists of one question, "What is your credence now for the proposition that our coin landed heads?"

Two popular solutions have been proposed: 1/3 and 1/2

The 1/3 solution

From wikipedia:

Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3.

Yes, it's true that only in a third of cases would heads precede her awakening.

Radford Neal (a statistician!) argues that 1/3 is the correct solution.

This [the 1/3] view can be reinforced by supposing that on each awakening Beauty is offered a bet in which she wins 2 dollars if the coin lands Tails and loses 3 dollars if it lands Heads. (We suppose that Beauty knows such a bet will always be offered.) Beauty would not accept this bet if she assigns probability 1/2 to Heads. If she assigns a probability of 1/3 to Heads, however, her expected gain is 2 × (2/3) − 3 × (1/3) = 1/3, so she will accept, and if the experiment is repeated many times, she will come out ahead.

Neal is correct (about the gambling problem).

These two arguments for the 1/3 solution appeal to intuition and make no obvious mathematical errors.   So why are they wrong?

Let's first start with probability laws and show why the 1/2 solution is correct. Just like with the Monty Hall problem, once you understand the solution, the wrong answer will no longer appeal to your intuition.

The 1/2 solution

P(Beauty woken up at least once| heads)=P(Beauty woken up at least once | tails)=1.  Because of the amnesia, all Beauty knows when she is woken up is that she has woken up at least once.  That event had the same probability of occurring under either coin outcome.  Thus, P(heads | Beauty woken up at least once)=1/2.  You can use Bayes' rule to see this if it's unclear.

Here's another way to look at it:

If it landed heads then Beauty is woken up on Monday with probability 1.

If it landed tails then Beauty is woken up on Monday and Tuesday.  From her perspective, these days are indistinguishable.  She doesn't know if she was woken up the day before, and she doesn't know if she'll be woken up the next day.  Thus, we can view Monday and Tuesday as exchangeable here.

A probability tree can help with the intuition (this is a probability tree corresponding to an arbitrary wake up day):

If Beauty was told the coin came up heads, then she'd know it was Monday.  If she was told the coin came up tails, then she'd think there is a 50% chance it's Monday and a 50% chance it's Tuesday.  Of course, when Beauty is woken up she is not told the result of the flip, but she can calculate the probability of each.

When she is woken up, she's somewhere on the second set of branches.  We have the following joint probabilities: P(heads, Monday)=1/2; P(heads, not Monday)=0; P(tails, Monday)=1/4; P(tails, Tuesday)=1/4; P(tails, not Monday or Tuesday)=0.  Thus, P(heads)=1/2.

Where the 1/3 arguments fail

The 1/3 argument says with heads there is 1 interview, with tails there are 2 interviews, and therefore the probability of heads is 1/3.  However, the argument would only hold if all 3 interview days were equally likely.  That's not the case here. (on a wake up day, heads&Monday is more likely than tails&Monday, for example).

Neal's argument fails because he changed the problem. "on each awakening Beauty is offered a bet in which she wins 2 dollars if the coin lands Tails and loses 3 dollars if it lands Heads."  In this scenario, she would make the bet twice if tails came up and once if heads came up.  That has nothing to do with probability about the event at a particular awakening.  The fact that she should take the bet doesn't imply that heads is less likely.  Beauty just knows that she'll win the bet twice if tails landed.  We double count for tails.

Imagine I said "if you guess heads and you're wrong nothing will happen, but if you guess tails and you're wrong I'll punch you in the stomach."  In that case, you will probably guess heads.  That doesn't mean your credence for heads is 1 -- it just means I added a greater penalty to the other option.

Consider changing the problem to something more extreme.  Here, we start with heads having probability 0.99 and tails having probability 0.01.  If heads comes up we wake Beauty up once.  If tails, we wake her up 100 times.  Thirder logic would go like this:  if we repeated the experiment 1000 times, we'd expect her woken up 990 after heads on Monday, 10 times after tails on Monday (day 1), 10 times after tails on Tues (day 2),...., 10 times after tails on day 100.  In other words, ~50% of the cases would heads precede her awakening. So the right answer for her to give is 1/2.

Of course, this would be absurd reasoning.  Beauty knows heads has a 99% chance initially.  But when she wakes up (which she was guaranteed to do regardless of whether heads or tails came up), she suddenly thinks they're equally likely?  What if we made it even more extreme and woke her up even more times on tails?

Implausible consequence of 1/2 solution?

Nick Bostrom presents the Extreme Sleeping Beauty problem:

This is like the original problem, except that here, if the coin falls tails, Beauty will be awakened on a million subsequent days. As before, she will be given an amnesia drug each time she is put to sleep that makes her forget any previous awakenings. When she awakes on Monday, what should be her credence in HEADS?

He argues:

The adherent of the 1/2 view will maintain that Beauty, upon awakening, should retain her credence of 1/2 in HEADS, but also that, upon being informed that it is Monday, she should become extremely confident in HEADS:
P+(HEADS) = 1,000,001/1,000,002

This consequence is itself quite implausible. It is, after all, rather gutsy to have credence 0.999999% in the proposition that an unobserved fair coin will fall heads.

It's correct that, upon awakening on Monday (and not knowing it's Monday), she should retain her credence of 1/2 in heads.

However, if she is informed it's Monday, it's unclear what she conclude.  Why was she informed it was Monday?  Consider two alternatives.

Disclosure process 1:  regardless of the result of the coin toss she will be informed it's Monday on Monday with probability 1

Under disclosure process 1, her credence of heads on Monday is still 1/2.

Disclosure process 2: if heads she'll be woken up and informed that it's Monday.  If tails, she'll be woken up on Monday and one million subsequent days, and only be told the specific day on one randomly selected day.

Under disclosure process 2, if she's informed it's Monday, her credence of heads is 1,000,001/1,000,002.  However, this is not implausible at all.  It's correct.  This statement is misleading: "It is, after all, rather gutsy to have credence 0.999999% in the proposition that an unobserved fair coin will fall heads."  Beauty isn't predicting what will happen on the flip of a coin, she's predicting what did happen after receiving strong evidence that it's heads.

ETA (5/9/2010 5:38AM)

If we want to replicate the situation 1000 times, we shouldn't end up with 1500 observations.  The correct way to replicate the awakening decision is to use the probability tree I included above. You'd end up with expected cell counts of 500, 250, 250, instead of 500, 500, 500.

Suppose at each awakening, we offer Beauty the following wager:  she'd lose $1.50 if heads but win $1 if tails.  She is asked for a decision on that wager at every awakening, but we only accept her last decision. Thus, if tails we'll accept her Tuesday decision (but won't tell her it's Tuesday). If her credence of heads is 1/3 at each awakening, then she should take the bet. If her credence of heads is 1/2 at each awakening, she shouldn't take the bet.  If we repeat the experiment many times, she'd be expected to lose money if she accepts the bet every time.

The problem with the logic that leads to the 1/3 solution is it counts twice under tails, but the question was about her credence at an awakening (interview).

ETA (5/10/2010 10:18PM ET)


Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3.

Another way to look at it:  the denominator is not a sum of mutually exclusive events.  Typically we use counts to estimate probabilities as follows:  the numerator is the number of times the event of interest occurred, and the denominator is the number of times that event could have occurred. 

For example, suppose Y can take values 1, 2 or 3 and follows a multinomial distribution with probabilities p1, p2 and p3=1-p1-p2, respectively.   If we generate n values of Y, we could estimate p1 by taking the ratio of #{Y=1}/(#{Y=1}+#{Y=2}+#{Y=3}). As n goes to infinity, the ratio will converge to p1.   Notice the events in the denominator are mutually exclusive and exhaustive.  The denominator is determined by n.

The thirder solution to the Sleeping Beauty problem has as its denominator sums of events that are not mutually exclusive.  The denominator is not determined by n.  For example, if we repeat it 1000 times, and we get 400 heads, our denominator would be 400+600+600=1600 (even though it was not possible to get 1600 heads!).  If we instead got 550 heads, our denominator would be 550+450+450=1450.  Our denominator is outcome dependent, where here the outcome is the occurrence of heads.  What does this ratio converge to as n goes to infinity?  I surely don't know.  But I do know it's not the posterior probability of heads.

Disconnect between Stated/Implemented Preferences

-4 Strange7 17 March 2010 02:26AM

Currently, the comment for which I've received the most positive karma by a factor of four is a joke about institutionalized ass-rape. A secondhand joke, effectively a quote with no source cited. Furthermore, the comment had, at best, tangential relevance to the subject of discussion. If anyone were to provide a detailed explanation of why they voted as they did, I predict that I would be appreciative.

Based on this evidence, which priors need to be adjusted? Discuss.

No One Knows What Science Doesn't Know

37 Eliezer_Yudkowsky 25 October 2007 11:47PM

At a family party some years ago, one of my uncles remarked on how little science really knows.  For example, we still have no idea how gravity works - why things fall down.

"Actually, we do know how gravity works," I said.  (My father, a Ph.D. physicist, was also present; but he wasn't even touching this one.)

"We do?" said my uncle.

"Yes," I said, "Gravity is the curvature of spacetime."  At this point I had still swallowed Feynman's line about being able to explain physics to one's grandmother, so I continued:  "You could say that the Earth goes around the Sun in a straight line.  Imagine a graph that shows both space and time, so that a straight line shows steady movement and a curved line shows acceleration.  Then curve the graph paper itself.  When you try to draw a straight line on the curved paper, you'll get what looks like acceleration -"

"I never heard about anything like that," said my uncle.

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Normal Cryonics

57 Eliezer_Yudkowsky 19 January 2010 07:08PM

I recently attended a small gathering whose purpose was to let young people signed up for cryonics meet older people signed up for cryonics - a matter of some concern to the old guard, for obvious reasons.

The young cryonicists' travel was subsidized.  I suspect this led to a greatly different selection filter than usually prevails at conferences of what Robin Hanson would call "contrarians".  At an ordinary conference of transhumanists - or libertarians, or atheists - you get activists who want to meet their own kind, strongly enough to pay conference fees and travel expenses.  This conference was just young people who took the action of signing up for cryonics, and who were willing to spend a couple of paid days in Florida meeting older cryonicists.

The gathering was 34% female, around half of whom were single, and a few kids.  This may sound normal enough, unless you've been to a lot of contrarian-cluster conferences, in which case you just spit coffee all over your computer screen and shouted "WHAT?"  I did sometimes hear "my husband persuaded me to sign up", but no more frequently than "I pursuaded my husband to sign up".  Around 25% of the people present were from the computer world, 25% from science, and 15% were doing something in music or entertainment - with possible overlap, since I'm working from a show of hands.

I was expecting there to be some nutcases in that room, people who'd signed up for cryonics for just the same reason they subscribed to homeopathy or astrology, i.e., that it sounded cool.  None of the younger cryonicists showed any sign of it.  There were a couple of older cryonicists who'd gone strange, but none of the young ones that I saw.  Only three hands went up that did not identify as atheist/agnostic, and I think those also might have all been old cryonicists.  (This is surprising enough to be worth explaining, considering the base rate of insanity versus sanity.  Maybe if you're into woo, there is so much more woo that is better optimized for being woo, that no one into woo would give cryonics a second glance.)

The part about actually signing up may also be key - that's probably a ten-to-one or worse filter among people who "get" cryonics.  (I put to Bill Faloon of the old guard that probably twice as many people had died while planning to sign up for cryonics eventually, than had actually been suspended; and he said "Way more than that.")  Actually signing up is an intense filter for Conscientiousness, since it's mildly tedious (requires multiple copies of papers signed and notarized with witnesses) and there's no peer pressure.

For whatever reason, those young cryonicists seemed really normal - except for one thing, which I'll get to tomorrow.  Except for that, then, they seemed like very ordinary people: the couples and the singles, the husbands and the wives and the kids, scientists and programmers and sound studio technicians.

It tears my heart out.

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An account of what I believe to be inconsistent behavior on the part of our editor

2 PeterS 17 December 2009 01:33AM

There was recently a submission here posing criticism a well-known contributor, Eliezer Yudkowsky.

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The Obesity Myth

12 Matt_Simpson 30 July 2009 12:12AM

Related To:  The Unfinished Mystery of the Shangri-La Diet and Missed Distinctions 

Megan McArdles blogs an interview with Paul Campos, author of The Obesity Myth.  I'll let anyone who is interest read the whole thing, but here's some interesting excerpts:

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Taking Occam Seriously

22 steven0461 29 May 2009 05:31PM

Paul Almond's site has many philosophically deep articles on theoretical rationality along LessWrongish assumptions, including but not limited to some great atheology, an attempt to solve the problem of arbitrary UTM choice, a possible anthropic explanation why space is 3D, a thorough defense of Occam's Razor, a lot of AI theory that I haven't tried to understand, and an attempt to explain what it means for minds to be implemented (related in approach to this and this).