# Probability is fake, frequency is real

Con­sider the Sleep­ing Beauty prob­lem. What do we mean by fair coin? It is meant that the coin will have 50-50 prob­a­bil­ity of heads or tails. But that is fake. It will ether come up heads or tail, be­cause the real world is de­ter­minis­tic. It is true that I don’t know the out­come. I don’t know if I am in a world of type “the coin will come up heads” or a world of type “the coin will come up tails”. But in this situ­a­tion I should be al­lowed to put what ever prior I want on the coins be­hav­ior.

Con­sider the Born rule of quan­tum me­chan­ics. If I mea­sure the spin of an elec­tron, then I will en­tan­gle the large ap­para­tus that is the my mea­sur­ing equip­ment with the spin of the elec­tron. We say that there are now two Everett branches, one where the ap­para­tus mea­sured spin up and one where the ap­para­tus mea­sured spin down. Be­fore I read of the re­sult, I don’t know which Hilbert branch I am in. I could be in ether, and I should be al­lowed to have what ever prior I want. So why the Born rule? Why to I do I be­lieve that the square am­pli­tude is the cor­rect way of as­sign­ing prob­a­bil­ity to which Hilbert branch I am in?

I be­lieve in the Born rule be­cause of the fre­quency of ex­per­i­men­tal out­comes in the past. The dis­tri­bu­tion of galax­ies in the sky can be traced back to the Born rule. I don’t have the gears on what is caus­ing the Born rule, but there are some­thing un­de­ni­ably real about galax­ies that trumps mere philo­soph­i­cal Bayesian ar­gu­ments about free­dom of pri­ors.

Imag­ine that you are offered a bet. Should you take it or not? There are sev­eral ar­gu­ment about what you should do in differ­ent situ­a­tions. For ex­am­ple, if you have finite amount of money, you should max­i­mize the E(log(money)) for each bet, (see e.g. Kelly crite­rion). How­ever, ev­ery such ar­gu­ment I have ever seen, is as­sum­ing that you will be con­fronted by a large num­ber of similar bets. This is be­cause prob­a­bil­ities only re­lay make sense if you sam­ple enough times from the ran­dom dis­tri­bu­tion you are con­sid­er­ing.

The no­tion of “fair coin” does not make sense if the coin is flipped only once. The right way to view the Sleep­ing Beauty prob­lem is to view it in it in the con­text of Re­peated Sleep­ing Beauty.

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• Seems you have a cat­e­gory er­ror, at least in the ti­tle. Prob­a­bil­ity is a model, fre­quency is an ob­ser­va­tion. A model can­not be fake, it can be a cer­tain de­gree of use­ful­ness in ex­plain­ing and pre­dict­ing the ob­ser­va­tions.

• What do we mean by fair coin? It is meant that the coin will have 50-50 prob­a­bil­ity of heads or tails. But that is fake. It will ether come up heads or tail, be­cause the real world is de­ter­minis­tic.

A small nit to pick per­haps, but it doesn’t fol­low that the world is de­ter­minis­tic be­cause the coin comes up ei­ther heads or tails. Per­haps the coin has liber­tar­ian free will and de­cides at the last mo­ment which side to come up, yet we’d make the same ob­ser­va­tion that the coin was ei­ther heads or tails in this case and would be wrong to say the world is de­ter­minis­tic. To make a claim about de­ter­minism you need to be say­ing some­thing about the fu­ture be­ing de­ter­mined (caused by) on the past. When we talk about fair coins and prob­a­bil­ities we are in­stead ex­press­ing some­thing about our un­cer­tainty about the fu­ture, not an ex­pres­sion of whether or not we think the fu­ture is de­ter­mined by the past.

• I don’t know what you mean by “should be al­lowed to put what­ever prior I want”. I mean, I guess no­body will stop you. But if your be­liefs are well ap­prox­i­mated by a par­tic­u­lar prior, then pre­tend­ing that they are ap­prox­i­mated by a differ­ent prior is go­ing to cause a mis­match be­tween your be­liefs and your be­liefs about your be­liefs.

[Nit­pick: The Kelly crite­rion as­sumes not only that you will be con­fronted with a large num­ber of similar bets, but also that you have some base level of risk-aver­sion (con­cave util­ity func­tion) that re­peated bets can smooth out into a log­a­r­ith­mic util­ity func­tion. If you start with a lin­ear util­ity func­tion then re­peat­ing the bets still gives you lin­ear util­ity, and the op­ti­mal strat­egy is to make ev­ery bet all-or-noth­ing when­ever you have an ad­van­tage. At least, this is true be­fore tak­ing into ac­count the re­source con­straints of the sys­tem you are bet­ting against.]

• I agree that “want” is not the cor­rect word ex­actly. What I mean by prior is an agents ac­tual a pri­ori be­liefs, so by defi­ni­tion there will be no mis-match there. I am not try­ing to say that you choose your prior ex­actly.

What I am ges­tur­ing at is that no prior is wrong, as long as it does not as­sign zero prob­a­bil­ity to the true out­come. And I think that much of the con­fu­sion in at­rophic situ­a­tion comes from try­ing to solve an un­der-con­strained sys­tem.

• We can talk about sin­gle-shot events, so long as we are al­lowed to in­clude multi-shot el­e­ments. For ex­am­ple, Sleep­ing Beauty in­cludes a coin. We can in­sist that Sleep­ing Beauty is only run once, but when we say the coin has 5050 chance of be­ing heads we are talk­ing about the long-run fre­quency.

• .the real world is deterministic

We don’t know that. Even if the way we quan­tify prob­a­bil­ity is a “map” fea­ture, prob­a­bil­ity is not just a set of quan­tities. 0.5 is not in­trin­si­cally a prob­a­bil­ity. Prob­a­bil­ity is the quan­tifi­ca­tion of some kind of in­de­ter­minism or lack of in­for­ma­tion. Un­der con­di­tions of perfect de­ter­minism and in­for­ma­tion, there is noth­ing for prob­a­bil­ity do do.

• I think you would be right if we lived in a clas­si­cal uni­verse. But given many wor­lds, there is a prin­ci­pled way in which a coin flip can be ran­dom, and a prin­ci­pled differ­ence be­tween flip­ping a coin and check­ing the trillionth digit of the dec­i­mal ex­pan­sion of .

Edit: I know you ac­knowl­edge this, but you don’t seem to draw the above con­clu­sion.