Capítulo 4 Análisis y discusión de los resultados
4.2 Análisis por categoría
4.2.3 Criterio vector 3, chacra 3
4.2.1
The Sleeping Beauty Problem as a Problem about Obser-
vation Bias
Are there, as Elga (2000) put it, “two sorts of uncertainty”—one “about what the world is like” and another one “about one’s own spatial and or temporal location in the world”— that are distinct in such a way that they require different handling in confirmation theory
4.2. Self-Locating Beliefs and Observation Bias 75
or epistemology more broadly? The Sleeping Beauty problem was brought back to the forefront by Elga (2000) in order to illustrate that they are distinct, and to show how the former bear on the latter. I will contend that the kind of information and uncertainty the Sleeping Beauty problem purports to deal with is in fact not specific to self-location, and that as a consequence it can’t be used to answer the questions it was intended to answer.
Here’s the standard setup of the problem: Beauty will be put to sleep for three days on Sunday night. Right after she falls asleep, a fair coin will be tossed to determine how many times she will be briefly woken. If the coin toss results in Heads, Beauty will be briefly woken only once, on Monday. If Tails, she will be woken twice: once on Monday, and once on Tuesday. But after each waking, Beauty will be put back to sleep with a drug that makes her completely forget about that waking. Now, if we ask Beauty, right after she has been woken but without telling her what day it is, what her credence should be that the coin came upHeads on Sunday night, what should her answer be?
Two answers are usually given to this question: “1
2” and “ 1
3”. For “halfers” (e.g., Lewis, 2001; White, 2006), that credence doesn’t depend on self-locating beliefs, and at no point did she learn anything that would alter her credence that the coin is fair and that there is a one in two chance that it landed Heads on Sunday night. For “thirders” (e.g., Elga, 2000; Dieks, 2007; Titelbaum, 2008), upon waking up in the middle of the experiment, Beauty’s credence should change merely because she finds herself in a dif- ferent situation than before the experiment. Without having any new information, her self-locating uncertainty makes her waking after Tails twice as likely as after Heads.
More precisely, according to Elga (2000), and with H: “the coin landed Heads,” T: “the coin landed Tails,” M: “it is now Monday,” U: “it is now Tuesday,” Beauty believes the coin to be fair and therefore should give equal credence toT&M and H&M. Moreover, because she’s not able to locate herself, she should give equal credence to T&M and T&U. Because H&M, T&M, and T&U are the only possible predicaments
Beauty could find herself in upon awaking, Elga asserted that the respective probabilities to be located in one of them sum up to 1. Consequently, Beauty’s credence that the coin landed Heads on Sunday if asked in the middle of the experiment should be 1
3. I will come back to Elga’s argument and its relevance for anthropic arguments later in §4.3.3. Thus we can see what role self-locating uncertainty should play for our beliefs about what the world is like according to thirders. I contend, however, that this isn’t what this problem illustrates. The canonical presentation of the Sleeping Beauty complicates it unnecessarily and obfuscates its meaning.4 Contrary to claims made by, e.g., Titelbaum (2013a, §9), when Beauty awakes in the middle of the experiment, she has not lost cer- tainty about herlocation as much as she has gained knowledge about her newly acquired
observation bias. In the middle of the experiment, she knows that she will be awoken and asked about her credence in H twice as often after aT toss than after aH toss. Outside of the experiment, she may assume that her questioner’s behavior won’t depend on the outcome of the coin toss. But by entering the experiment, she acquires information about her questioner’s asymmetric behavior, rather than losing certainty about her temporal location.
The Sleeping Beauty problem can indeed be reformulated as one about observation bias, in which self-locating uncertainty or memory loss play no role. Consider that, instead of having to use fictitious memory-loss-inducing sleeping pills, Beauty takes part in the following experiment, being fully aware of its setting: a quizzer is sitting at a table behind a screen, with his head above the screen; he regularly throws a fair coin, which then falls completely silently on a shock-absorbing mat; he asks Beauty what side she thinks the coin last landed, but he asks that question twice in a row without throwing the coin again each time it landsTails. The problem now is to know what Beauty should answer in the middle of that experiment. Like in the standard version of the problem, she never knows if the question she answers follows a Heads toss or a Tails toss.
4Moreover, one might argue that doing so by invoking gratuitously a woman being asked to take memory-erasing pills is inconsiderate.
4.2. Self-Locating Beliefs and Observation Bias 77
Beauty will on average achieve a better ratio of correct answers (and even a perfect one in an ideal situation) if she chooses to answer “Heads” only a third of the time. Therefore, if she wants to increase her chance of giving a correct answer to the quizzer, Beauty should indeed modify her answers when she is in the middle of the experiment. She shouldn’t do so because she doesn’t know where she is as much as because she has been made aware of the bias (observer bias or sampling bias) of the quizzer for whom
Tails tosses count twice as much.
In the standard problem and this alternate experiment alike, if she didn’t know of that bias, and if she were able to keep a tally of her correct guesses, Beauty would conclude that if the coin is fair, then the quizzer is biased (or vice versa). She could for instance start to suspect that the quizzer is cheating or can’t see half the Heads results. But if Beauty is cognizant of her questioner’s bias and if in the long run she concludes that one third of the answers should be “Heads”, this would corroborate her belief that the coin is fair.
This problem, however, is nothing special for confirmation theory. Observation bias in any measurement is handled in a similar manner. However we choose to present the Sleep- ing Beauty problem, neither memories nor location—let alone self-locating uncertainty— play a distinctive role such that it requires a special handling, distinct from usual evi- dential reasoning.5
4.2.2
Beauty’s Bets in a Rigged Game
Bradley and Leitgeb (2006); Cisewski et al. (2015), following earlier work by Seiden- feld et al. (1990), have argued that the Sleeping Beauty problem is an example where her credence and her betting behavior don’t match. Neither memories, uncertainty nor indifference about one’s location need be assumed in this betting framework either.
According to de Finetti (1974), the probability that an agent assigns to an event E
can be elicited by asking how much she is willing to bet that E, knowing that she would earn $1 if E and $0 otherwise. That price, p, is theelicited probability of E. If p is the price of the gamble andxthe probability ofE, the expected utility of the gamble isx−p. A fair bet is one where the agent expects neither gain nor loss (p=x).
Consider the alternative version of the Sleeping Beauty problem previously intro- duced, with the questioner hidden behind a screen and regularly asking Beauty how much she is willing to bet that the last coin toss wasHeads, but asking her twice as often after each Tails toss. In this situation, for Beauty’s bet to be fair to her and the bookie (i.e., the questioner), she has to account for the fact that in the long run, she will lose her wager twice as often as she will receive $1. If she believes the coin to be fair (i.e., x= 2), the only way that neither she nor the bookie wins in the long run is if her wager is $1
3. See Appendix 4.A for details.
In either version of the Sleeping Beauty problem (i.e., the standard version or the one with a coin tossed behind a screen and where no sleep is involved), her betting price is equivalent to her credence in Heads before observation bias is corrected for. In other words, her betting price is the elicited probability of a biased event, i.e., her credence about a skewed sample.