Paradoxes in all anthropic probabilities

In a pre­vi­ous post, I re-dis­cov­ered full non-in­dex­i­cal up­dat­ing (FNC), an an­thropic the­ory I’m ashamed to say I had once known and then for­got. Thanks to Wei Dai for re­mind­ing me of that.

There is a prob­lem with FNC, though. In fact, there are prob­lems with all an­thropic prob­a­bil­ity the­o­ries. Both FNC and SIA vi­o­late con­ser­va­tion of ex­pected ev­i­dence: you can be in a situ­a­tion where you know with cer­tainty that your fu­ture prob­a­bil­ity will be differ­ent in a par­tic­u­lar di­rec­tion, from your cur­rent one. SSA has a differ­ent prob­lem: it al­lows you to make de­ci­sions that change the prob­a­bil­ity of past events.

Th­ese para­doxes are pre­sented to illus­trate the fact that an­thropic prob­a­bil­ity is not a co­her­ent con­cept, and that deal­ing with mul­ti­ple copies of a sin­gle agent is in the realm of de­ci­sion the­ory.

FNC and ev­i­dence non-conservation

Let’s pre­sume that the band­width of the hu­man brain is bits per minute. Then we flip a coin. Upon it com­ing up heads, we cre­ate iden­ti­cal copies of you. Upon it com­ing up tails, we cre­ate copies of you.

Then if we as­sume that the ex­pe­riences of your differ­ent copies are ran­dom, for the first minute, you will give equal prob­a­bil­ity to heads and tails. That’s be­cause there is a be­ing with ex­actly the same ob­ser­va­tions as you, in both uni­verses.

After two min­utes, you will shift to odds in favour of tails: you’re cer­tain there’s a be­ing with your ob­ser­va­tions in the tails uni­verse, and, with prob­a­bil­ity , there’s one in the heads uni­verse.

After a full three min­utes, you will fi­nally sta­bil­ise on odds in favour of tails, and stay there.

Thus, dur­ing the first minute, you know that FNC will be giv­ing you differ­ent odds in the com­ing min­utes, and you can pre­dict the di­rec­tion those odds will take.

If the ob­ser­va­tions are non-ran­dom, then the di­ver­gence will be slower, and the FNC odds will be chang­ing for a longer pe­riod.

SIA and ev­i­dence non-conservation

If we use SIA in­stead of FNC, then, in the above situ­a­tion, the odds of tails will be and will stay there, so that set­ting is not an is­sue for SIA.

To show a prob­lem with SIA, as­sume there is one copy of you, that we flip a coin, and, if comes out tails, we will im­me­di­ately du­pli­cate you (putting the du­pli­cate in a sep­a­rate room). If it comes out heads, we will wait a minute be­fore du­pli­cat­ing you.

Then SIA im­plies in favour of tails dur­ing that minute, but equal odds af­ter­wards.

You can’t get around this with tweaked refer­ences classes: one of the good prop­er­ties of SIA is that it works the same what­ever the refer­ence class, as long as it in­cludes agent cur­rently sub­jec­tively in­dis­t­in­guish­able from you.

SSA and chang­ing the past

SSA has a lot of is­sues. It has the whole prob­lem with refer­ence classes; these are hard to define co­her­ently, and agents in differ­ent refer­ence classes with the same pri­ors can agree to dis­agree (for in­stance, if we ex­pect that there will be a sin­gle gen­der in the fu­ture, then if I’m in the refer­ence class of males, I ex­pect that sin­gle gen­der will be fe­male—and the op­po­site will be ex­pected for some­one in the refer­ence class of fe­males). It vi­o­lates causal­ity: it as­signs differ­ent prob­a­bil­ities to an event, purely de­pend­ing on the fu­ture con­se­quence of that event.

But I think I’ll fo­cus on an­other way it vi­o­lates causal­ity: your cur­rent ac­tions can change the prob­a­bil­ity of past events.

Sup­pose that the prover­bial coin is flipped, and that if it comes up heads, one ver­sion of you is cre­ated, and, if it comes up tails, copies of you are cre­ated. You are the last of these copies: ei­ther the only one in the heads world, or the last one in the tails world, you don’t know. Un­der SSA, you as­sign odds of in favour of heads.

You have a con­ve­nient lever, how­ever. If you pull it, then fu­ture copies of you will be cre­ated, in the heads world only (noth­ing will hap­pen in the tails world). There­fore, of you pull it, the odds of the coin be­ing tails—an event long past, and known to be past—will shift to from to in favour.

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