The Absolute Self-Selection Assumption

There are many con­fused dis­cus­sions of an­thropic rea­son­ing, both on LW and in sur­pris­ingly main­stream liter­a­ture. In this ar­ti­cle I will dis­cuss UDASSA, a frame­work for an­thropic rea­son­ing due to Wei Dai. This frame­work has se­ri­ous short­com­ings, but at pre­sent it is the only one I know which pro­duces rea­son­able an­swers to rea­son­able ques­tions; at the mo­ment it is the only frame­work which I would feel com­fortable us­ing to make a real de­ci­sion.

I will dis­cuss 3 prob­lems:

1. In an in­finite uni­verse, there are in­finitely many copies of you (in­finitely many of which are Boltz­mann brains). How do you as­sign a mea­sure to the copies of your­self when the uniform dis­tri­bu­tion is un­available? Do you rule out spa­tially or tem­po­rally in­finite uni­verses for this rea­son?

2. Naive an­throp­ics ig­nore the sub­strate on which a simu­la­tion is run­ning and count how many in­stances of a simu­lated ex­pe­rience ex­ist (or how many dis­tinct ver­sions of that ex­pe­rience ex­ist). Th­ese be­liefs are in­con­sis­tent with ba­sic in­tu­itions about con­scious ex­pe­rience, so we have to aban­don some­thing in­tu­itive.

3. The Born prob­a­bil­ities seem mys­te­ri­ous. They can be ex­plained (as well as any law of physics can be ex­plained) by UDASSA.

Why An­thropic Rea­son­ing?

When I am try­ing to act in my own self-in­ter­est, I do not know with cer­tainty the con­se­quences of any par­tic­u­lar de­ci­sion. I com­pare prob­a­bil­ity dis­tri­bu­tions over out­comes: an ac­tion may lead to one out­come with prob­a­bil­ity 12, and a differ­ent out­come with prob­a­bil­ity 12. My brain has prefer­ences be­tween prob­a­bil­ity dis­tri­bu­tions built into it.

My brain is not built with the ma­chin­ery to de­cide be­tween differ­ent uni­verses each of which con­tains many simu­la­tions I care about. My brain can’t even re­ally grasp the no­tion of differ­ent copies of me, ex­cept by first con­vert­ing to the lan­guage of prob­a­bil­ity dis­tri­bu­tions. If I am fac­ing the prospect of be­ing copied, the only way I can grap­ple with it is by rea­son­ing “I have a 50% chance of re­main­ing me, and a 50% chance of be­com­ing my copy.” After think­ing in this way, I can hope to in­tel­li­gently trade-off one copy’s prefer­ences against the other’s us­ing the same ma­chin­ery which al­lows me to make de­ci­sions with un­cer­tain out­comes.

In or­der to perform this rea­son­ing in gen­eral, I need a bet­ter frame­work for an­thropic rea­son­ing. What I want is a prob­a­bil­ity dis­tri­bu­tion over all pos­si­ble ex­pe­riences (or “ob­server-mo­ments”), so that I can use my ex­ist­ing prefer­ences to make in­tel­li­gent de­ci­sions in a uni­verse with more than one ob­server I care about.

I am go­ing to leave many ques­tions un­re­solved. I don’t un­der­stand con­ti­nu­ity of ex­pe­rience or iden­tity, so I am sim­ply not go­ing to try to be self­ish (I don’t know how). I don’t un­der­stand what con­sti­tutes con­scious ex­pe­rience, so I am not go­ing to try and ex­plain it. I have to rely on a com­plex­ity prior, which in­volves an un­ac­cept­able ar­bi­trary choice of a no­tion of com­plex­ity.

The Ab­solute Self-Selec­tion Assumption

A thinker us­ing Solomonoff in­duc­tion searches for the sim­plest ex­pla­na­tion for its own ex­pe­riences. It even­tu­ally learns that the sim­plest ex­pla­na­tion for its ex­pe­riences is the de­scrip­tion of an ex­ter­nal lawful uni­verse in which its sense or­gans are em­bed­ded and a de­scrip­tion of that em­bed­ding.

As hu­mans us­ing Solomonoff in­duc­tion, we go on to ar­gue that this ex­ter­nal lawful uni­verse is real, and that our con­scious ex­pe­rience is a con­se­quence of the ex­is­tence of cer­tain sub­struc­ture in that uni­verse. The ab­solute self-se­lec­tion as­sump­tion dis­cards this ad­di­tional step. Rather than sup­pos­ing that the prob­a­bil­ity of a cer­tain uni­verse de­pends on the com­plex­ity of that uni­verse, it takes as a prim­i­tive ob­ject a prob­a­bil­ity dis­tri­bu­tion over pos­si­ble ex­pe­riences.

By the same rea­son­ing that led a nor­mal Solomonoff in­duc­tor to ac­cept the ex­is­tence of an ex­ter­nal uni­verse as the best ex­pla­na­tion for its ex­pe­riences, the least com­plex de­scrip­tion of your con­scious ex­pe­rience is the de­scrip­tion of an ex­ter­nal lawful uni­verse and di­rec­tions for find­ing the sub­struc­ture em­body­ing your ex­pe­rience within that sub­struc­ture.

This re­quires spec­i­fy­ing a no­tion of com­plex­ity. I will choose a uni­ver­sal com­putable dis­tri­bu­tion over strings for now, to mimic con­ven­tional Solomonoff in­duc­tion as closely as pos­si­ble (and be­cause I know noth­ing bet­ter). The re­sult­ing the­ory is called UDASSA, for Univer­sal Distri­bu­tion + ASSA.

Re­cov­er­ing In­tu­itive Anthropics

Sup­pose I cre­ate a perfect copy of my­self. In­tu­itively, I would like to weight the two copies equally. Similarly, my an­thropic no­tion of “prob­a­bil­ity of an ex­pe­rience” should match up with my in­tu­itive no­tion of prob­a­bil­ity. For­tu­nately, UDASSA re­cov­ers in­tu­itive an­throp­ics in in­tu­itive situ­a­tions.

The short­est de­scrip­tion of me is a pair (U, x), where U is a de­scrip­tion of my uni­verse and x is a de­scrip­tion of where to find me in that uni­verse. If there are two copies of me in the uni­verse, then the ex­pe­rience of each can be de­scribed in the same way: (U, x1) and (U, x2) are de­scrip­tions of ap­prox­i­mately equal com­plex­ity, so I weight the ex­pe­rience of each copy equally. The to­tal ex­pe­rience of my copies is weighted twice as much as the to­tal ex­pe­rience of an un­copied in­di­vi­d­ual.

Part of x is a de­scrip­tion of how to nav­i­gate the ran­dom­ness of the uni­verse. For ex­am­ple, if the last (truly ran­dom) coin I saw flipped came up heads, then in or­der to spec­ify my ex­pe­riences you need to spec­ify the re­sult of that coin flip. An equal num­ber of equally com­plex de­scrip­tions point to the ver­sion of me who saw heads and the ver­sion of me who saw tails.

Prob­lem #1: In­finite Cosmologies

Modern physics is con­sis­tent with in­finite uni­verses. An in­finite uni­verse con­tains in­finitely many ob­servers (in­finitely many of which share all of your ex­pe­riences so far), and it is no longer sen­si­ble to talk about the “uniform dis­tri­bu­tion” over all of them. You could imag­ine tak­ing a limit over larger and larger vol­umes, but there is no par­tic­u­lar rea­son to sus­pect such a limit would con­verge in a mean­ingful sense. One solu­tion that has been sug­gested is to choose an ar­bi­trary but very large vol­ume of space­time, and to use a uniform dis­tri­bu­tion over ob­servers within it. Another solu­tion is to con­clude that in­finite uni­verses can’t ex­ist. Both of these ex­pla­na­tions are un­satis­fac­tory.

UDASSA pro­vides a differ­ent solu­tion. The prob­a­bil­ity of an ex­pe­rience de­pends ex­po­nen­tially on the com­plex­ity of spec­i­fy­ing it. Just ex­ist­ing in an in­finite uni­verse with a short de­scrip­tion does not guaran­tee that you your­self have a short de­scrip­tion; you need to spec­ify a po­si­tion within that in­finite uni­verse. For ex­am­ple, if your ex­pe­riences oc­cur 34908172349823478132239471230912349726323948123123991230 steps af­ter some nat­u­rally speci­fied time 0, then the (some­what lengthy) de­scrip­tion of that time is nec­es­sary to de­scribe your ex­pe­riences. Thus the to­tal mea­sure of all ob­server-mo­ments within a uni­verse is finite.

Prob­lem #2: Split­ting Simulations

Con­sider a com­puter which is 2 atoms thick run­ning a simu­la­tion of you. Sup­pose this com­puter can be di­vided down the mid­dle into two 1 atom thick com­put­ers which would both run the same simu­la­tion in­de­pen­dently. We are faced with an un­for­tu­nate di­chotomy: ei­ther the 2 atom thick simu­la­tion has the same weight as two 1 atom thick simu­la­tions put to­gether, or it doesn’t.

In the first case, we have to ac­cept that some com­puter simu­la­tions count for more, even if they are run­ning the same simu­la­tion (or we have to de-du­pli­cate the set of all ex­pe­riences, which leads to se­ri­ous prob­lems with Boltz­mann ma­chines). In this case, we are faced with the prob­lem of com­par­ing differ­ent sub­strates, and it seems im­pos­si­ble not to make ar­bi­trary choices.

In the sec­ond case, we have to ac­cept that the op­er­a­tion of di­vid­ing the 2 atom thick com­puter has moral value, which is even worse. Where ex­actly does the tran­si­tion oc­cur? What if each layer of the 2 atom thick com­puter can run in­de­pen­dently be­fore split­ting? Is phys­i­cal con­tact re­ally sig­nifi­cant? What about com­put­ers that aren’t phys­i­cally co­her­ent? What two 1 atom thick com­put­ers pe­ri­od­i­cally syn­chro­nize them­selves and self-de­struct if they aren’t syn­chro­nized: does this syn­chro­niza­tion effec­tively de­stroy one of the copies? I know of no way to ac­cept this pos­si­bil­ity with­out ex­tremely counter-in­tu­itive con­se­quences.

UDASSA im­plies that simu­la­tions on the 2 atom thick com­puter count for twice as much as simu­la­tions on the 1 atom thick com­puter, be­cause they are eas­ier to spec­ify. Given a de­scrip­tion of one of the 1 atom thick com­put­ers, then there are two de­scrip­tions of equal com­plex­ity that point to the simu­la­tion run­ning on the 2 atom thick com­puter: one de­scrip­tion point­ing to each layer of the 2 atom thick com­puter. When a 2 atom thick com­puter splits, the to­tal num­ber of de­scrip­tions point­ing to the ex­pe­rience it is simu­lat­ing doesn’t change.

Prob­lem #3: The Born Probabilities

A quan­tum me­chan­i­cal state can be de­scribed as a lin­ear com­bi­na­tion of “clas­si­cal” con­figu­ra­tions. For some rea­son we ap­pear to ex­pe­rience our­selves as be­ing in one of these clas­si­cal con­figu­ra­tions with prob­a­bil­ity pro­por­tional the co­effi­cient of that con­figu­ra­tion squared. Th­ese prob­a­bil­ities are called the Born prob­a­bil­ities, and are some­times de­scribed ei­ther as a se­ri­ous prob­lem for MWI or as an un­re­solved mys­tery of the uni­verse.

What hap­pens if we ap­ply UDASSA to a quan­tum uni­verse? For one, the ex­is­tence of an ob­server within the uni­verse doesn’t say any­thing about con­scious ex­pe­rience. We need to spec­ify an al­gorithm for ex­tract­ing a de­scrip­tion of that ob­server from a de­scrip­tion of the uni­verse.

Con­sider the ran­dom­ized al­gorithm A: com­pute the state of the uni­verse at time t, then sam­ple a clas­si­cal con­figu­ra­tion with prob­a­bil­ity pro­por­tional to its squared in­ner product with the uni­ver­sal wave­func­tion.

Con­sider the ran­dom­ized al­gorithm B: com­pute the state of the uni­verse at time t, then sam­ple a clas­si­cal con­figu­ra­tion with prob­a­bil­ity pro­por­tional to its in­ner product with the uni­ver­sal wave­func­tion.

Us­ing ei­ther A or B, we can de­scribe a sin­gle ex­pe­rience by spec­i­fy­ing a ran­dom seed, and pick­ing out that ex­pe­rience within the clas­si­cal con­figu­ra­tion out­put by A or B us­ing that ran­dom seed. If this is the short­est ex­pla­na­tion of an ex­pe­rience, the prob­a­bil­ity of an ex­pe­rience is pro­por­tional to the num­ber of ran­dom seeds which pro­duce clas­si­cal con­figu­ra­tions con­tain­ing it.

The uni­verse as we know it is typ­i­cal for an out­put of A but com­pletely im­prob­a­ble as an out­put of B. For ex­am­ple, the ob­served be­hav­ior of stars is con­sis­tent with al­most all ob­ser­va­tions weighted ac­cord­ing to al­gorithm A, but with al­most no ob­ser­va­tions weighted ac­cord­ing to al­gorithm B. Al­gorithm A con­sti­tutes an im­mensely bet­ter de­scrip­tion of our ex­pe­riences, in the same sense that quan­tum me­chan­ics con­sti­tutes an im­mensely bet­ter de­scrip­tion of our ex­pe­riences than clas­si­cal physics.

You could also imag­ine an al­gorithm C, which uses the same se­lec­tion as al­gorithm B to point to the Everett branch con­tain­ing a physi­cist about to do an ex­per­i­ment, but then uses al­gorithm A to de­scribe the ex­pe­riences of the physi­cist af­ter do­ing that ex­per­i­ment. This is a hor­ribly com­plex way to spec­ify an ex­pe­rience, how­ever, for ex­actly the same rea­son that a Solomonoff in­duc­tor places very low prob­a­bil­ity on the laws of physics sud­denly chang­ing for just this one ex­per­i­ment.

Of course this leaves open the ques­tion of “why the Born prob­a­bil­ities and not some other rule?” Al­gorithm B is a valid way of spec­i­fy­ing ob­servers, though they would look ex­actly as for­eign as ob­serves with differ­ent rules of physics (Wei Dai has sug­gested that the struc­tures speci­fied by al­gorithm B are not even self-aware as jus­tifi­ca­tion for the Born rule). The fact that we are de­scribed by al­gorithm A rather than B is no more or less mys­te­ri­ous than the fact that the laws of physics are like so in­stead of some other way.

In the same way that we can retroac­tively jus­tify our laws of physics by ap­peal­ing to their el­e­gance and sim­plic­ity (in a sense we don’t yet re­ally un­der­stand) I sus­pect that we can jus­tify se­lec­tion ac­cord­ing to al­gorithm A rather than al­gorithm B. In an in­finite uni­verse, al­gorithm B doesn’t even work (be­cause the sum of the in­ner prod­ucts of the uni­ver­sal wave­func­tion with the clas­si­cal con­figu­ra­tions is in­finite) and even in a finite uni­verse al­gorithm B nec­es­sar­ily in­volves the ad­di­tional step of nor­mal­iz­ing the prob­a­bil­ity dis­tri­bu­tion or else pro­duc­ing non­sense. More­over, al­gorithm A is a nicer math­e­mat­i­cal ob­ject than al­gorithm B when the evolu­tion of the wave­func­tion is uni­tary, and so the same con­sid­er­a­tions that sug­gest el­e­gant laws of physics sug­gest al­gorithm A over B (or some other al­ter­na­tive).

Note that this is not the core of my ex­pla­na­tion of the Born prob­a­bil­ities; in UDASSA, choos­ing a se­lec­tion pro­ce­dure is just as im­por­tant as de­scribing the uni­verse, and so some ex­plicit sort of ob­server se­lec­tion is a nec­es­sary part of the laws of physics. We pre­dict the Born rule to hold in the fu­ture be­cause it has held in the past, just like we ex­pect the laws of physics to hold in the fu­ture be­cause they have held in the past.

In sum­mary, if you use Solomonoff in­duc­tion to pre­dict what you will see next based on ev­ery­thing you have seen so far, your pre­dic­tions about the fu­ture will be con­sis­tent with the Born prob­a­bil­ities. You only get in trou­ble when you use Solomonoff in­duc­tion to pre­dict what the uni­verse con­tains, and then get bogged down in the ques­tion “Given that the uni­verse con­tains all of these ob­servers, which one should I ex­pect to be me?”