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.

Next: Re­peated (and im­proved) Sleep­ing Beauty problem

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