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].