Adam Elga

Self‐locating belief and the Sleeping Beauty problem

The Sleeping Beauty problem:1 Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking.2 When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads? 1So named by Robert Stalnaker (who first learned of examples of this kind in unpublished work by Arnold Zuboff). This problem appears as Example 5 of Piccione 1997, which motivates two distinct answers but suspends judgment as to which answer is correct (1997:12–14). Aumann 1997 uses a fair lottery approach to analyse a similar problem. Adapted to the Sleeping Beauty problem, that analysis yields the same answer as the one I will defend in section 2. However, unlike the argument in Aumann 1997, my argument does not depend on betting considerations. 2The precise effect of the drug is to reset your belief-state to what it was just before you were put to sleep at the beginning of the experiment. If the existence of such a drug seems fanciful, note that it is possible to pose the problem without it — all that matters is that the person put to sleep believes that the setup is as I have described it.

A model of the rodent head direction system

Head direction cells in the postsubiculum (PoS, also known as dorsal presubiculum) were first described by Ranck et al. [10]. In subsequent work, Taube et al. [14] characterized these cells as having triangular tuning curves: the firing rate drops off linearly from a peak at the preferred direction until it reaches a baseline value. Taube et al. [15] report that PoS cells typically have baseline-to-baseline tuning curve widths of 100°. Similar cells have been found in the anterior thalamic nuclei (ATN) [4, 6, 13]. See Figure 1(b) for a sample PoS tuning curve. These curves can also be modeled very closely by Gaussians with an average standard deviation of approximately 66° [4, 18].