(This is the second post in a short sequence discussing evidence and arguments presented by Christopher Ryan and Cacilda Jethá's Sex at Dawn, inspired by the spirit of Kaj_Sotala's recent discussion of What Intelligence Tests Miss. It covers Part II: Lust in Paradise and Part III: The Way We Weren't.)

Forager anthropology is a discipline that is easy to abuse. It relies on unreliable first-hand observations of easily misunderstood cultures that are frequently influenced by the presence of modern observers. These cultures are often exterminated or assimilated within decades of their discovery, making it difficult to confirm controversial claims and discoveries. But modern-day foraging societies are the most direct source of evidence we have about our pre-agricultural ancestors; in many ways, they are agriculture's control group, living in conditions substantially similar to the ones under which our species evolved. The standard narrative of human sexual evolution ignores or manipulates the findings of forager anthropology to support its claims, and this is no doubt responsible for much of its confused support.

Steven Pinker is one of the most prominent and well-respected advocates of the standard narrative, both on Less Wrong and elsewhere. Eliezer has referenced him as an authority on evolutionary psychology. One commenter on the first post in this series claimed that Pinker is "the only mainstream academic I'm aware of who visibly demonstrates the full suite of traditional rationalist virtues in essentially all of his writing." Another cited Pinker's claim that 20-60% of hunter-gatherer males were victims of lethal human violence ("murdered") as justification for a Malthusian view of human nature. 

That 20-60% number comes from a claim about war casualties in a 2007 TED talk Pinker gave on "the myth of violence", for which he drew upon several important findings in forager anthropology. (The talk is based on an argument presented in the third chapter of The Blank Slate; there is a text version of the talk available, but it omits the material on forager anthropology that Ryan and Jethá critique.)

At 2:45 in the video Pinker displays a slide which reads

Until 10,000 years ago, humans lived as hunter-gatherers, without permanent settlements or government.

He also points out that modern hunter-gatherers are our best evidence for drawing conclusions about those prehistoric hunter-gatherers; in both these statements he is in accordance with nearly universal historical, anthropological, and archaeological opinion. Pinker's next slide is a chart from The Blank Slate, originally based on the research of Lawrence Keeley. Sort of. It is labeled as "the percentage of male deaths due to warfare," with bars for eight hunter-gatherer societies that range from approximately 15-60%. The problem is that of these eight cultures, zero are migratory hunter-gatherers.

(This post is the beginning of a short sequence discussing evidence and arguments presented by Christopher Ryan and Cacilda Jethá's Sex at Dawn, inspired by the spirit of Kaj_Sotala's recent discussion of What Intelligence Tests Miss. It covers Part I: On the Origin of the Specious.)

Sex at Dawn: The Prehistoric Origins of Modern Sexuality was first brought to my attention by a rhapsodic mention in Dan Savage's advice column, and while it seemed quite relevant to my interests I am generally very skeptical of claims based on evolutionary psychology. I did eventually decide to pick up the book, primarily so that I could raid its bibliography for material for an upcoming post on jealousy management, and secondarily to test my vulnerability to confirmation bias. I succeeded in the first and failed in the second: Sex at Dawn is by leaps and bounds the best evolutionary psychology book I've read, largely because it provides copious evidence for its claims.¹ I mention the strength of my opinion as a disclaimer of sorts, so that careful readers may take the appropriate precautions.

The book's first section focuses on the current generally accepted explanation for human sexual evolution, which the authors call "the standard narrative." It's an explanation that should be quite familiar to regular LessWrong readers: men are attracted to fertile-appearing women and try to prevent them from having sex with other men so as to confirm the paternity of their offspring; women are attracted to men who seem like they will be good providers for their children and try to prevent them from forming intimate bonds with other women so as to maintain access to their resources.

When we choose behavior, including verbal behavior, it's sometimes tempting to do what is most likely to be right without paying attention to how costly it is to be wrong in various ways or looking for a safer alternative.

If you've taken a lot of standardized tests, you know that some of them penalize guessing and some don't.  That is, leaving a question blank might be better than getting a wrong answer, or they might have the same result.  If they're the same, of course you guess, because it can't hurt and may help.  If they take off points for wrong answers, then there's some optimal threshold at which a well-calibrated test-taker will answer.  For instance, the ability to rule out one of four choices on a one-point question where a wrong answer costs a quarter point means that you should guess from the remaining three - the expected point value of this guess is positive.  If you can rule out one of four choices and a wrong answer costs half a point, leave it blank.

If you have ever asked a woman who wasn't pregnant when the baby was due, you might have noticed that life penalizes guessing.

If you're risk-neutral, you still can't just do whatever has the highest chance of being right; you must also consider the cost of being wrong.  You will probably win a bet that says a fair six-sided die will come up on a number greater than 2.  But you shouldn't buy this bet for a dollar if the payoff is only $1.10, even though that purchase can be summarized as "you will probably gain ten cents".  That bet is better than a similarly-priced, similarly-paid bet on the opposite outcome; but it's not good.

There's a few factors at work to make guessing tempting anyway:

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.