The Fermi Paradox: What did Sandberg, Drexler and Ord Really Dissolve?

(Cross-posted from my blog)

So this pa­per by the trio from the FHI, An­ders Sand­berg, Eric Drexler and Toby Ord (SDO for short) has been talked about quite a bit, on LessWrong, on SSC and on Red­dit. It is about how their Monte-Carlo calcu­la­tions based on prob­a­bil­ity dis­tri­bu­tions rather than on the usual point es­ti­mates of the Drake equa­tion ap­par­ently dis­solves the ques­tion of why we are seem­ingly alone in the Uni­verse that is sup­posed to be teem­ing with in­tel­li­gent life, if one takes the Coper­ni­can idea “we are not spe­cial” se­ri­ously. One grim sug­ges­tion is that there is a Great Filter that is still in front of us and is al­most guaran­teed to kill us off be­fore the hu­man civ­i­liza­tion reaches the tech­nolog­i­cal lev­els ob­serv­able by other civ­i­liza­tions like ours.

There are plenty of other ideas, most ad­dress­ing var­i­ous fac­tors in the Drake equa­tion, but the one ad­vanced in SDO is quite differ­ent from the main­stream: they say that es­ti­mat­ing these fac­tors is a wrong way to go, be­cause the un­cer­tainty in these very small prob­a­bil­ities is so large, the point es­ti­mates are all but mean­ingless. In­stead they sug­gest that the cor­rect ap­proach is some­thing along the fol­low­ing lines: First, as­sume a rea­son­able prob­a­bil­ity dis­tri­bu­tion for each fac­tor, then draw a value for each fac­tor based on their prob­a­bil­ity dis­tri­bu­tions, calcu­late the re­sult­ing ex­pected value of the num­ber of cur­rently de­tectable civ­i­liza­tions, then re­peat this pro­cess many times to cre­ate a syn­thetic prob­a­bil­ity dis­tri­bu­tion of the num­ber of this civ­i­liza­tions, and fi­nally ex­tract the odds of us be­ing alone in the uni­verse from this dis­tri­bu­tion. And that is what they did, and con­cluded that the odds of us be­ing alone in the Milky Way are some­thing like 1:3. Thus, ac­cord­ing to SDO, there is no para­dox, an av­er­age uni­verse is nat­u­rally a des­o­late place.

Their to the Fermi para­dox solu­tion is basically

Due to ran­dom chance, some of the pa­ram­e­ters in the Drake equa­tion, we do not know which, we do not know why, are many or­ders of mag­ni­tude smaller in our uni­verse than pre­vi­ously es­ti­mated.

I have an is­sue with call­ing it “dis­solv­ing the para­dox”, since it doesn’t an­swer any of the prac­ti­cal ques­tions about the uni­verse we live in. Fermi’s ques­tion, “Where is ev­ery­body?” re­mains open.

But I may have mi­s­un­der­stood the pa­per. So, what fol­lows is an at­tempt to un­der­stand their logic by re­pro­duc­ing it. Here I will an­a­lyze their toy model, as it has all the salient fea­tures lead­ing to their con­clu­sion:

There are nine pa­ram­e­ters (f1, f2, . . .) mul­ti­plied to­gether to give the prob­a­bil­ity of ETI [Ex­tra-ter­res­trial in­tel­li­gence] aris­ing at each star. Sup­pose that our true state of knowl­edge is that each pa­ram­e­ter could lie any­where in the in­ter­val [0, 0.2], with our un­cer­tainty be­ing uniform across this in­ter­val, and be­ing un­cor­re­lated be­tween pa­ram­e­ters.

A point es­ti­mate would be tak­ing a mean for each fac­tor and mul­ti­ply­ing them, giv­ing one-in-a-billion chance per star, which re­sults in a vir­tual cer­tainty of ETI at least some­where in a galaxy of hun­dreds of billions of stars.

Let’s try a dis­tri­bu­tion es­ti­mate in­stead: take 9 ran­dom num­bers from a uniform dis­tri­bu­tion over the in­ter­val [0, 0.2]. Here is a bunch of sam­ple runs, the ex­pected val­ues of ETIs in the toy galaxy, and the odds of the toy galaxy be­ing empty:

No­tice that, out of the 10 sam­ple galax­ies in this ex­am­ple, about half show con­sid­er­able odds of be­ing empty, high­light­ing the differ­ence be­tween us­ing a dis­tri­bu­tion and the point es­ti­mates. SDO likely had run a lot more simu­la­tions to get more ac­cu­racy, and I did, too. Here is the dis­tri­bu­tion of the odds for a given num­ber of ETIs in the galaxy, based on about ten mil­lion runs:

The hori­zon­tal axis is the num­ber of ETIs, and the ver­ti­cal axis is the prob­a­bil­ity of this num­ber of ETIs to hap­pen in a given toy galaxy drawn at ran­dom, given the prob­a­bil­ity dis­tri­bu­tion for each fac­tor, as speci­fied above.

No­tice how wide this dis­tri­bu­tion is: some galax­ies are com­pletely bereft of ETIs, while oth­ers have hun­dreds and even thou­sands of them!

The above graph is very close to the power law: the prob­a­bil­ity of the num­ber of ETIs goes roughly as the num­ber of ETIs to the power of −1.2. This means that the es­ti­mate of the ex­pected num­ber of ETIs is ac­tu­ally di­ver­gent, though the me­dian num­ber of ETI’s per galaxy is finite and about 30. So, some­what para­dox­i­cally,

Us­ing the dis­tri­bu­tional es­ti­mate in­stead of a point es­ti­mate both in­creases the likely num­ber of ETIs per galaxy, and the frac­tion of empty galax­ies.

The pa­per states that “Monte Carlo simu­la­tion shows that this ac­tu­ally pro­duces an empty galaxy 21.45% of the time,” but does not spec­ify the way this num­ber was calcu­lated. From the plot above, the prob­a­bil­ity of hav­ing be­tween zero and one ETIs, what­ever it might mean, is 24.63%. A differ­ent way of calcu­lat­ing the av­er­age odds of an empty toy galaxy would be to calcu­late the odds of each sam­ple galaxy to be empty as

P(empty) = (1-p_1*p_2*...*p_9)^(10^11)

then av­er­age all these odds. This ap­proach gives the frac­tion of empty galax­ies as 21.43%, very close to the num­ber quoted in the pa­per.

This toy ex­am­ple demon­strates nicely the main re­sult of the SDO pa­per: the usual point es­ti­mate pro­duces a wildly in­ac­cu­rate ex­pected frac­tion of galax­ies with no ETIs in them. The rest of the SDO pa­per is fo­cused on calcu­lat­ing a more re­al­is­tic ex­am­ple, based on the cur­rent best guesses for the dis­tri­bu­tions of each fac­tor in the Drake equa­tion:

Us­ing these dis­tri­bu­tions and their fur­ther re­fine­ments to calcu­late the odds yield the fol­low­ing con­clu­sion:

When we up­date this prior in light of the Fermi ob­ser­va­tion, we find a
sub­stan­tial prob­a­bil­ity that we are alone in our galaxy, and per­haps even in our
ob­serv­able uni­verse (53%–99.6% and 39%–85% re­spec­tively). ’Where are they?’
— prob­a­bly ex­tremely far away, and quite pos­si­bly be­yond the cos­molog­i­cal
hori­zon and for­ever un­reach­able.

If our uni­verse is drawn at ran­dom from the pool of fac­tors with the dis­tri­bu­tions as sug­gested by the pa­per, then there are sub­stan­tial odds that we are alone in the ob­serv­able uni­verse.

This is be­cause some of the fac­tors in the Drake equa­tion, due to their un­cer­tainty, can end up many or­ders of mag­ni­tude smaller than the value used for a point es­ti­mate, the one where this un­cer­tainty is not taken into ac­count. There is no claim which of the pa­ram­e­ters are that small, since they differ for differ­ent des­o­late uni­verses. Just the (bad) luck of the draw.

So, the Fermi para­dox has been solved, right? Well, yes and no. Once we draw a set of pa­ram­e­ters to use in the Drake equa­tion in one spe­cific uni­verse, for ex­am­ple ours, we are still left with the task of ex­plain­ing their val­ues. We are no closer to un­der­stand­ing the Great Filter, if any, than be­fore. Is abio­ge­n­e­sis ex­tremely rare? Do ETIs self-de­struct quickly? Are there some spe­cial cir­cum­stances re­quired for life to thrive be­yond the planet be­ing in the Goldilocks zone? Is there the sin­gle­ton effect where the first civ­i­liza­tion out of the gate takes over the galaxy? Who knows. SDO con­cludes that

This re­sult dis­solves the Fermi para­dox, and in do­ing so re­moves any need to in­voke spec­u­la­tive mechanisms by which civ­i­liza­tions would in­evitably fail to have ob­serv­able effects upon the uni­verse.

I find that this con­clu­sion does not fol­low from the main re­sult of the pa­per. We live in this one uni­verse, and we are stuck with the spe­cific set of val­ues of the fac­tors in the Drake equa­tion that our uni­verse hap­pened to have. It is quite pos­si­ble that in our uni­verse there is a Great Filter “by which civ­i­liza­tions would in­evitably fail to have ob­serv­able effects upon the uni­verse,” be­cause, for ex­am­ple one spe­cific pa­ram­e­ter has the value that is many or­ders of mag­ni­tude lower than the es­ti­mate, and it would be re­ally use­ful to know which one and why. The er­ror SDO is mak­ing is look­ing at the dis­tri­bu­tion of the uni­verses, whereas the Fermi para­dox ap­plies to the one we are stuck with. A real re­s­olu­tion of the para­dox would be, for ex­am­ple, de­ter­min­ing which pa­ram­e­ters in the Drake equa­tion are van­ish­ingly low and why, not sim­ply declar­ing that it is ex­ceed­ingly likely that a ran­domly drawn uni­verse has one or sev­eral van­ish­ingly small pa­ram­e­ters lead­ing to it be­ing very likely bereft of ETIs.