# In SIA, reference classes (almost) don’t matter

This is an­other write-up of a fact that is gen­er­ally known, but that I haven’t seen proven ex­plic­itly: the fact that SIA does not de­pend upon the refer­ence class.

Speci­fi­cally:

• As­sume there are a finite num­ber of pos­si­ble uni­verses . Let be a refer­ence class of finitely many agents in those uni­verses, and as­sume you are in . Let be the refer­ence class of agents sub­jec­tively in­dis­t­in­guish­able from you. Then SIA us­ing is in­de­pen­dent of as long as .

Proof:

Let be a set of uni­verses for some in­dex­ing set , and a prob­a­bil­ity dis­tri­bu­tion over them. For a uni­verse , let be the num­ber of agents in the refer­ence class in .

Then if is the prob­a­bil­ity dis­tri­bu­tion from SIA us­ing :

• .

We now wish to up­date on our own sub­jec­tive ex­pe­rience . Since there are agents in our refer­ence class, and have sub­jec­tively in­dis­t­in­guish­able ex­pe­riences, this up­dates the prob­a­bil­ities by weights , which is just . After nor­mal­is­ing, this is:

Thus this ex­pres­sion is in­de­pen­dent of .

Given some mea­sure the­ory (and mea­sure the­o­retic re­stric­tions on to make sure ex­pres­sions like con­verge), the re­sult ex­tends to in­finite classes of uni­verses, with in the proof in­stead of .

• When you calcu­late pR(Ui|sub), you perform the fol­low­ing trans­for­ma­tion pR(Ui)→pR(Ui)×R0(Ui)/​R(Ui), but an R(Ui) seems to go miss­ing. Can any­one ex­plain?

• Where does the R(Ui) go miss­ing? It’s there in the sub­se­quent equa­tion.

• pR(Ui) already had an R(Ui), then you di­vided by it, but the origi­nal fac­tor dis­ap­pears so you are left with a di­vided by R(Ui). But I don’t see where the origi­nal fac­tor of R(Ui) went, which would have re­sulted in can­cel­ling.

• You are cor­rect, I dropped a in the proof, thanks! Put it back in, and the proof is now shorter.

• As I un­der­stand from above, in SIA the real refer­ence class is “the class of ob­servers who is sub­jec­tively in­dis­t­in­guish­able from me” - and that is why SIA doesn’t de­pend on any other refer­ence classes which I could be a mem­ber. How­ever, it doesn’t ex­clude the use of SSA logic for SSA-re­lated con­clu­sions.

An ex­am­ple of SSA logic: I am a mem­ber of a class of peo­ple who was born be­tween equa­tor and a pole of Earth, and by the fact of my birth I was ran­domly se­lected from this class. Thus, the place of my birth should be rather ran­domly (but ac­count­ing for differ­ent pop­u­la­tion den­si­ties) se­lected be­tween equa­tor and pole, and un­likely to be ex­actly on the equa­tor or on the pole. I was born at 55 lat­i­tude, so SSA logic work in pre­dict­ing my lat­i­tude of birth.

I could be a mem­ber of many differ­ent SSA-classes and for each of them make in­de­pen­dent pre­dic­tions about my po­si­tion in them.

For SIA the class of “sub­jec­tively in­dis­t­in­guish­able” my copies could be also not very ex­act. Differ­ent in­ter­pre­ta­tion of such class is:

1) ev­ery­body is me who have the same thought pro­cess as me now. There could be a lot of them, even on Earth.

2) ev­ery­body, who has the to­tal sum of all vi­sual (and other) ex­pe­riences as me, even de­spite the fact that I will not be able to ac­count for all differ­ences as they are too small to ac­count.

3) ev­ery­body who has ex­actly the same brain as me. This class hun­dred or­ders of mag­ni­tude more rare than (2), as the same ex­pe­rience could be gen­er­ated by differ­ent brains.

I think that “true” SIA class is some­where be­tween (1) and (2) - or more likely, there is no “true SIA class”, the same way as there is no true SSA-class, and differ­ent types of SIA could be used to an­swer differ­ent ques­tions.

• and differ­ent types of SIA could be used to an­swer differ­ent ques­tions.

Yep. ^_^

• This re­sult seems strange to me, even though the maths seems to check out. Is there a con­cep­tual ex­pla­na­tion of why this should be the case?

• Maybe: larger refer­ence classes make the uni­verses more likely, but make it less likely that you would be a spe­cific mem­ber of that refer­ence class, so when you up­date on who you are in the class, the two effects can­cel out.

More con­cep­tu­ally: in SAI, the defi­ni­tion of refer­ence class com­mutes with re­stric­tions on that refer­ence class. So it doesn’t mat­ter if you take the refer­ence class of all hu­mans, then spe­cial­ise to the ones al­ive to­day, then spe­cial­ise to you; or take the refer­ence class of all hu­mans al­ive to­day, then spe­cial­ise to you; or just take the refer­ence class of you. SIA is, in a sense, sen­si­ble with re­spect to up­dat­ing.

Does that help?

• Thanks, that’s helpful. Ac­tu­ally, now that you’ve put it that way, I re­call hav­ing known this fact at some point in the past.

• Another way of see­ing SAI + up­date on your­self: weigh each uni­verse by the ex­pected num­ber of ex­act (sub­jec­tive) copies of you in them, then renor­mal­ise.

• Yes, but note that SSA can get this same re­sult. All they have to do is say that their refer­ence class is R—what­ever set the SIA per­son uses, they use the same set. If they make this move, then they are refer­ence-class-in­de­pen­dent to ex­actly the same de­gree as SIA.

• SSA is not refer­ence class in­de­pen­dent. If it uses , then the SSA prob is (rather that ), which is , which is not in­de­pen­dent of (con­sider dou­bling the size of in one world only—that makes that world less likely rel­a­tive to all the oth­ers).

• Ah, my mis­take, sorry. I was think­ing of a differ­ent defi­ni­tion of refer­ence-class-in­de­pen­dent than you were; I should have read more closely.

• Oh, what defi­ni­tion were you us­ing? Any­thing in­ter­est­ing? (or do you mean be­fore up­dat­ing on your own ex­pe­riences?)

• Some­times when peo­ple say SIA is refer­ence-class in­de­pen­dent & SSA isn’t, they mean it as an ar­gu­ment that SIA is bet­ter than SSA, be­cause it is philo­soph­i­cally less prob­le­matic: The choice of refer­ence class is ar­bi­trary, so if we don’t have to make that choice, our the­ory is over­all more el­e­gant. This was the sort of thing I had in mind.

On that defi­ni­tion, SSA is only more ar­bi­trary than SIA if it makes the refer­ence class differ­ent from the class of all ob­servers. (Which some pro­po­nents of SSA have done) SIA has a con­cept of ob­server too, at least, a con­cept of ob­server-in­dis­t­in­guish­able-from-me (which pre­sum­ably is proper sub­set of ob­server, though now that I think about it this might be challenged. Maybe I was dou­bly wrong—maybe SIA only needs the con­cept of ob­server-in­dis­t­in­guish­able-from-me).