Insights from the randomness/​ignorance model are genuine

(Based on the ran­dom­ness/​ig­no­rance model pro­posed in 1 2 3.)

The bold claim of this se­quence thus far is that the ran­dom­ness/​ig­no­rance model solves a sig­nifi­cant part of the an­throp­ics puz­zle. (Not ev­ery­thing since it’s still in­com­plete.) In this post I ar­gue that this “solu­tion” is gen­uine, i.e. it does more than just re­define terms. In par­tic­u­lar, I ar­gue that my defi­ni­tion of prob­a­bil­ity for ran­dom­ness is the only rea­son­able choice.

The only ax­iom I need for this claim is that prob­a­bil­ity must be con­sis­tent with bet­ting odds in all cases: if comes true in two of three situ­a­tions where is ob­served, and this is known, then needs to be , and no other an­swer is ac­cept­able. This idea isn’t new; the prob­lem with it is that it doesn’t ac­tu­ally pro­duce a defi­ni­tion of prob­a­bil­ity, be­cause we might not know how of­ten comes true if is ob­served. It can­not define prob­a­bil­ity in the origi­nal Pre­sump­tu­ous Philoso­pher prob­lem, for ex­am­ple.

But in the con­text of the ran­dom­ness/​ig­no­rance model, the ap­proach be­comes ap­pli­ca­ble. Stat­ing my defi­ni­tion for when un­cer­tainty is ran­dom in one sen­tence, we get

Your un­cer­tainty about , given ob­ser­va­tion , is ran­dom iff you know the rel­a­tive fre­quency with which hap­pens, eval­u­ated across all ob­ser­va­tions that, for you, are in­dis­t­in­guish­able to with re­gard to .

Where “rel­a­tive fre­quency” is the fre­quency of com­pared to , i.e. you know that hap­pens in out of cases. A good look at this defi­ni­tion shows that it is pre­cisely the con­di­tion needed to ap­ply the bet­ting odds crite­rion. So the model sim­ply di­vides ev­ery­thing into those cases where you can ap­ply bet­ting odds and those where you can’t.

If the Sleep­ing Beauty ex­per­i­ment is re­peated suffi­ciently of­ten us­ing a fair coin, then roughly half of all ex­per­i­ments will run in the 1-in­ter­view ver­sion, and the other half will run the 2-in­ter­view ver­sion. In that case, Sleep­ing Beauty’s un­cer­tainty is ran­dom and the rea­son­ing from 3 goes through to out­put for it be­ing Mon­day. The ex­per­i­ment be­ing re­peated suffi­ciently of­ten might be con­sid­ered a rea­son­ably mild re­stric­tion; in par­tic­u­lar, it is a given if the uni­verse is large enough that ev­ery­thing which ap­pears once ap­pears many times. Given that Sleep­ing Beauty is still con­tro­ver­sial, the model must thus be ei­ther non­triv­ial or wrong, hence “gen­uine”.

Here is an al­ter­na­tive jus­tifi­ca­tion for my defi­ni­tion of ran­dom prob­a­bil­ity. Sup­pose is the hy­poth­e­sis we want to eval­u­ate (like “to­day is Mon­day”) and is the full set of ob­ser­va­tions we cur­rently have (for­mally, the full brain state of Sleep­ing Beauty). Then what we care about is the value of . Now con­sider the term ; let’s call it . If is known, then can be com­puted as , so knowl­edge of im­plies knowl­edge of and vice-versa. But is more “fun­da­men­tal” than , in the sense that it can be defined as the ra­tio of two fre­quen­cies. Take all situ­a­tions in which – or any other a set of ob­ser­va­tions which, from your per­spec­tive, is in­dis­t­in­guish­able to – is ob­served, and count in how many of those is true vs. false. The ra­tio of these two val­ues is .

A look at the above crite­rion for ran­dom­ness shows that it’s just an­other way of say­ing that the value of is known. Since, again, the value of de­ter­mines the value of , this means that the defi­ni­tion of prob­a­bil­ity as bet­ting odds, in the case that the rele­vant un­cer­tainty is ran­dom, falls al­most di­rectly out of the for­mula.