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 .