Repeated (and im­proved) Sleep­ing Beauty problem

Fol­low up to: Prob­ab­il­ity is fake, fre­quency is real

There is some­thing wrong with the nor­mal for­mu­la­tion of the Sleep­ing Beauty prob­lem. More pre­cisely, there is some­thing wrong about pos­tu­lat­ing a single “fair” ran­dom coin flip. So here is an im­proved ver­sion of the Sleep­ing Beauty prob­lem. After ex­plain­ing the setup, I will re­cover the nor­mal Sleep­ing Beauty prob­lem, but in a more well defined way.

There are no truly ran­dom coins. There are only pseudo ran­dom coins which has the prop­erty that you don’t have the ca­pa­city to cal­cu­late the out­come. A fair pseudo ran­dom coin have the ad­di­tional prop­erty that when flipped enough times, the ra­tio of Heads v.s. Tails will ap­proach one. Note that fair­ness is only defined if you ac­tu­ally flip the coin a suf­fi­cient num­ber of times. Be­cause of this, the Sleep­ing Beauty prob­lem should be a re­peated game.

(Al­tern­at­ively, you could solve this by us­ing coun­ter­fac­tu­als. However, we don’t yet know how to deal with coun­ter­fac­tu­als. Also, I sus­pect that any method of hand­ling coun­ter­fac­tu­als will be, at best, use­ful but wrong.)

Repeated Sleep­ing Beauty setup: Every Sunday a mys­ter­i­ous per­son flips a pseudo ran­dom fair coin. If the coin comes up Heads, Sleep­ing Beauty will wake up on Monday, and then sleep for the rest of the week. If the coin comes up Tails, she will wake up on Monday and Tues­day and then sleep for the rest of the week. No-one is telling Sleep­ing Beauty what is go­ing on, she gets to rely on her own past ex­per­i­ences.

Every morn­ing when Sleep­ing Beauty wakes up she does not know what day it is. However there is an easy ex­per­i­ment she can do to find out, namely ask­ing any­one she meets on the street. Be­cause Sleep­ing is a curi­ous per­son, she is keep­ing a sci­ence journal. Every day she finds out what day it is and writes it down. She soon no­tices some pat­terns.

1) There are two kinds of days, Monday and Tues­day.

2) Every Tues­day is fol­lowed by a Monday.

3) A Monday can be fol­lowed by either a Tues­day or a Monday.

After some more time she starts to no­tice the fre­quen­cies of which dif­fer­ent days oc­cur.

13 of days are Mondays that are fol­lowed by Monday (cor­res­ponds to Heads & Monday)

13 of days are Mondays that are fol­lowed by Tues­day (cor­res­ponds to Tails & Monday)

13 of days are Tues­days (cor­res­ponds to Tails & Tues­day)

Sleep­ing Beauty tries to find more pat­terns in the data, but none of the more com­plic­ated hy­po­thes­izes she can come up with sur­vives fur­ther ob­ser­va­tion.

Re­cov­er­ing the ori­ginal Sleep­ing Beauty prob­lem: Sleep­ing Beauty have been slack­ing off for a few days, and not ask­ing for what day it was. What like­li­hood should she as­sign to the cur­rent day be­ing a Monday fol­lowed by Monday?

The ob­vi­ous an­swer based on Sleep­ing Beauty’s own ex­per­i­ence is 13.

Con­clu­sion and after-though: If you take your prob­ab­il­ity from how things have played out in the past, you will learn the Thirder po­s­i­tion /​ Self-in­dic­a­tion as­sump­tion (SIA). Also, do­ing what has worked well in the past leads to Eviden­tial de­cision the­ory (EDT). This is a sad fact of the uni­verse, be­cause EDT com­bined with SIA leads to a sort of double count­ing of ac­tions which add up to the wrong policy [cita­tion].