Bayesian analysis of the astrobiological

Bayesian analysis of the astrobiological implications of life's early emergence on Earth David S. Spiegel . Edwin L. Turner t I 'Institute for Advanced Study. Pnnceton. NJ 00540.1 Dept. of Astrophysical Sciences. Princeton Univ.. Princeton. NJ 08544. USA. and :Institute for the Physics and Mathematics of the Universe. The Univ. of Tokyo, Kashiwa 2274568. Japan Submitted to Proceedings of the National Academy of Sciences of the United States of America arXiv:1107.3835v4 [astro-ph.EP] 13 Apr 2012 Life arose on Earth sometime in the first few hundred million years after the young planet had cooled to the point that it could support water-based organisms on its surface. The early emergence of life on Earth has been taken as evidence that the probability of abiogen- esis is high. if starting from young-Earth-like conditions. We revisit this argument quantitatively in a Bayesian statistical framework. By constructing a simple model of the probability of abiogenesis. we calculate a Bayesian estimate of its posterior probability, given the data that life emerged fairly early in Earth's history and that. billions of years later. curious creatures noted this fact and considered its implications. We find that. given only this very limited empirical information, the choice of Bayesian prior for the abiogenesis proba- bility parameter has a dominant influence on the computed posterior probability. Although terrestrial life's early emergence provides evi- dence that life might be common in the Universe if early-Earth-like conditions are. the evidence is inconclusive and indeed is consistent with an arbitrarily low intrinsic probability of abiogenesis for plausible uninformative priors. Finding a single case of life arising indepen- dently of our lineage (on Earth. elsewhere in the Solar System. or on an extrasolar planet) would provide much stronger evidence that abiogenesis is not extremely rare in the Universe. Astrobiology Abbreviations: Gr. gigayear (10" years); PDF, probability density function; CDF, cumulative distribution function Introduction Astrobiology is fundamentally concerned with whether ex- traterrestrial life exists and, if so, how abundant it is in the Universe. The most direct and promising approach to answer- ing these questions is surely empirical, the search for life on other bodies in the Solar System [1, 2] and beyond in other planetary systems [3, 4]. Nevertheless, a theoretical approach is possible in principle and could provide a useful complement to the more direct lines of investigation. In particular, if we knew the probability per unit time and per unit volume of abiogenesis in a pre-biotic environ- ment as a function of its physical and chemical conditions and if we could determine or estimate the prevalence of such environments in the Universe, we could make a statistical esti- mate of the abundance of extraterrestrial life. This relatively straightforward approach is, of course, thwarted by our great ignorance regarding both inputs to the argument at present. There does, however, appear to be one possible way of fi- nessing our lack of detailed knowledge concerning both the process of abiogenesis and the occurrence of suitable pre- biotic environments (whatever they might be) in the Universe. Namely, we can try to use our knowledge that life arose at least once in an environment (whatever it was) on the early Earth to try to infer something about the probability per unit time of abiogenesis on an Earth-like planet without the need (or ability) to say how Earth-like it need be or in what ways. We will hereinafter refer to this probability per unit time, which can also be considered a rate, as A or simply the "probability of abiogenesis." Any inferences about the probability of life arising (given the conditions present on the early Earth) must be informed by how long it took for the first living creatures to evolve. By definition, improbable events generally happen infrequently. It follows that the duration between events provides a metric (however imperfect) of the probability or rate of the events. The time-span between when Earth achieved pre-biotic condi- tions suitable for abiogenesis plus generally habitable climatic conditions Is, 6, 7] and when life first arose, therefore, seems to serve as a basis for estimating A. Revisiting and quantifying this analysis is the subject of this paper. We note several previous quantitative attempts to address this issue in the literature, of which one [8] found, as we do, that early abiogenesis is consistent with life being rare, and the other [9] found that Earth's early abiogenesis points strongly to life being common on Earth-like planets (we com- pare our approach to the problem to that of [9] below, in- cluding our significantly different results).' lAirthermore, an argument of this general sort has been widely used in a qual- itative and even intuitive way to conclude that A is unlikely to be extremely small because it would then be surprising for abiogenesis to have occurred as quickly as it did on Earth [12, 13, I4, 15, 16, 17, 18]. Indeed, the early emergence of life on Earth is often taken as significant supporting evidence for "optimism" about the existence of extra-terrestrial life (i.e., for the view that it is fairly common) (19, 20, 9]. The major motivation of this paper is to determine the quantitative va- lidity of this inference. We emphasize that our goal is not to derive an optimum estimate of A based on all of the many lines of available evidence, but simply to evaluate the implication of life's early emergence on Earth for the value of A. A Bayesian Formulation of the Calculation Bayes's theorem [21] can be wrii Ion as P[MID] = (11DIMNPpno,.[MD/P[D]. Here. we take M to be a model and V to be data. In order to us• this equation to evalu- ate the posterior probability of abiogenesis, we must specify appropriate M and D. t daverkisedu Reserved for Publication Footnotes There am two unpublished works (1101 and (II)). of which we became ware after wbenil,ion of this papa. that also conclude that early Ilk as Earth does not rule out the FO5bl•ty that abiogenesis is improbable www.pnas.orgbegi/eki/10.1073/pnas.0703640104 PNAS l Issue Date I Volume I Issue Number 1 1-11 EFTA01071746 A Poisson or Uniform Rate Model. In considering the devel- opment of life on a planet. we suggest that a reasonable, if simplistic, model is that it is a Poisson process during a pe- riod of time from foam until tmax. In this model, the conditions on a young planet preclude the development of life for a time period of Irmo after its formation. Furthermore, if the planet remains lifeless until t„,„ has elapsed, it will remain lifeless thereafter as well because conditions no longer permit life to arise. For a planet around a solar-type star, tnax is almost certainly 10 Gyr (10 billion years, the main sequence life- time of the Sun) and could easily be a substantially shorter period of time if there is something about the conditions on a young planet that are necessary for abiogenesis. Between these limiting times, we posit that there is a certain probabil- ity per unit time (A) of life developing. For train < t < Luau: then, the probability of life arising n times in time t is P[A, n, = Ppobt. = C AO -I mm) {AO !min)]" [1] where t is the time since the formation of the planet. This formulation could well be questioned on a number of grounds. Perhaps most fundamentally, it treats abiogenesis as though it were a single instantaneous event and implicitly assumes that it can occur in only a single way (i.e., by only a single process or mechanism) and only in one type of physical environment. It is, of course, far more plausible that abiogen- esis is actually the result of a complex chain of events that take place over some substantial period of time and perhaps via different pathways and in different environments. How- ever, knowledge of the actual origin of life on Earth, to say nothing of other possible ways in which it might originate, is so limited that a more complex model is not yet justified. In essence, the simple Poisson event model used in this paper attempts to "integrate out" all such details and treat abio- genesis as a "black box" process: certain chemical and phys- ical conditions as input produce a certain probability of life emerging as an output. Another Sue is that A, the probabil- ity per unit time, could itself be a function of time. In fact, the claim that life could not have arisen outside the window (tni„,tro.) is tantamount to saying that A = 0 for t ≤ !min and for t ≥ tmax. Instead of switching from 0 to a fixed value instantaneously, A could exhibit a complicated variation with time. If so, however, P[A,n,fi is not represented by the Pois- son distribution and eq. (1) is not valid. Unless a particular (non top-hat-function) tune-variation of A is suggested on the- oretical grounds, it seems unwise to add such unconstrained complexity. A further criticism is that A could be a function of n: it could be that life arising once (or more) changes the probabil- ity per unit time of life arising again. Since we are primarily interested in the probability of life arising at all - i.e., the probability of n 0 0 - we can define A simply to be the value appropriate for a prebiotic planet (whatever that value may be) and remain agnostic as to whether it differs for n ≥ 1. Thus, within the adopted model, the probability of life aris- ing is one minus the probability of it not arising: Plife = I — PPotsson EA, 0,t] = 1 n• C —Mt . [2] A Minimum Evolutionary Time Constraint. Naively, the single datum informing our calculation of the posterior of A appears to be simply that life arose on Earth at least once, approxi- mately 3.8 billion years ago (give or take a few hundred million years). There is additional significant context for this datum, however. Recall that the standard claim is that, since life arose early on the only habitable planet that we have exam- ined for inhabitants, the probability of abiogenesis is proba- Models of to = 4.5 Cyr-Old Planets o e thetical Conserv.' Consery .2 tialsci.c tnti„ 0.5 0.5 0.5 0.5 temorge 0.51 1.3 1.3 0.7 tmax 10 1.4 10 10 Sterols. 1 2 3.1 1 trequired 3.5 1.4 1.4 3.5 All 0.01 0.80 0.80 0.20 a:2 3.00 0.90 0.90 3.00 300 1.1 1.1 15 All times are in Cyr. Two "Conservative" (Conserv.) models are shown, to indicate that !required may be limited either by a small value of /max ("Conserv. i"), or by a large value of ote„3,.. ("Conserv.2"). bly high (in our language, A is probably large). This stan- dard argument neglects a potentially important selection ef- fect, namely: On Earth, it took nearly 4 Gyr for evolution to lead to organisms capable of pondering the probability of life elsewhere in the Universe. If this is a necessary duration, then it would be impossible for us to find ourselves on, for example, a (-4.5-Gyr old) planet on which life first arose only after the passage of 3.5 billion years (221. On such planets there would not yet have been enough time for creatures capable of such contemplations to evolve. In other words, if evolution requires 3.5 Gyr for life to evolve from the simplest forms to intelligent, questioning beings, then we had to find ourselves on a planet where life arose relatively early, regardless of the value of A. In order to introduce this constraint into the calculation we define 8t„.„4, as the minimum amount of time required af- ter the emergence of life for cosmologically curious creatures to evolve, tom,ngo as the age of the Earth from when the earliest extant evidence of life remains (though life might have actu- ally emerged earlier), and to as the current age of the Earth. The data, then, are that life arose on Earth at least once, ap- proximately 3.8 billion years ago, and that this emergence was early enough that human beings had the opportunity subse- quently to evolve and to wonder about their origins and the possibility of life elsewhere in the Universe. In equation form, !emerge < to — Otevolvo• The Likelihood Term. We now seek to evaluate the P[DIM] term in Bayes's theorem. Let ta,,,oi„d a min[to — &evoke, Gnash Our existence on Earth requires that life ap- peared within !required. In other words, t„„,„fr od is the max- imum age that the Earth could have had at the origin of life in order for humanity to have a chance of showing up by the present. We define Se to be the set of all Earth-like worlds of age approximately to in a large, unbiased volume and L[1] to be the subset of St on which life has emerged within a time t. Litrecperoal is the set of planets on which life emerged early enough that creatures curious about abio- genesis could have evolved before the present (to), and, pre- suming te=„ = < tpmo,„d (which we know was the case for Earth), glemargo] is the subset of Wrequiradl on which life emerged as quickly as it did on Earth. Correspondingly, Nst, NG, and NL, are the respective numbers of planets in sets Se, L[trequirea], and gtomorgel. The fractions sot, a Mr/Nse 2 An °hematite nuy to derive equation (3) is to let E = "abiocenSs occurred between emi r, and and .R —']begins occurred between Ism and I mpor.d " We then have. from 2 I worre.pnas.erdegi/doi/10.10T3/pnas.0709640104 Spiegel & Turner EFTA01071747 and cote a Nie/N,s, are, respectively, the fraction of Earth- like planets on which life arose within ty,c,„;„d and the frac- tion on which life emerged within t,„,„,o. The ratio r a co‘b,ot, = Ne,,,Wer is the fraction of Lt, on which life arose as soon as it (lid on Earth. Given that we had to find our- selves on such a planet in the set Ltr in order to write and read about this topic, the ratio r characterizes the probability of the data given the model if the probability of intelligent observers arising is independent of the time of abiogenesis (so long as abiogenesis occurs before tre„,„i„d). (This last as- sumption might seem strange or unwarranted, but the effect of relaxing this assumption is to make it more likely that we would find ourselves on a planet with early abiogenesis and therefore to reduce our limited ability to infer anything about A from our observations.) Since co‘ = I— PpoinorP, 0, immerge] and <At =1 - Proh,..4A,O,t,„,qui,,il, we may write that p[, I A4] I onterge Imln)] [3] 1— exp[ exp[ —A(4.„,vd„d — /min)] if ,min < Leman,. < ta..quired (and P[DI.A4] = 0 otherwise). This is called the "likelihood function," and represents the proba- bility of the observation(s), given a particular model. It is via this function that the data "condition" our prior beliefs about A in standard Bayesian terminology. Limiting Behavior of the Likelihood. It is instructive to con- sider the behavior of equation (3) in some interesting limits. Fbr — Gem) C 1. the numerator and denominator of equation (3) each go approximately as the argument of the exponential function; therefore, in this limit, the likelihood function is approximately constant: PEDIAll 4 !emerge — tram [4] ...required — train This result is intuitively easy to understand as follows: If A is sufficiently small, it is overwhelmingly likely that abiogene- sis occurred only once in the history of the Earth, and by the assumptions of our model, the one event is equally likely to oc- cur at any time during the interval between train and troquired. The chance that this will occur by t,,,, o is then just the fraction of that total interval that has passed by !mange - the result given in equation (4). In the other limit, when A(t0moexe - train) >, 1, the numer- ator and denominator of equation (3) are both approximately I. In this case, the likelihood function is also approximately constant (and equal to unity). This result is even more in- tuitively obvious since a very large value of A implies that abiogenesis events occur at a high rate (given suitable condi- tions) and are thus likely to have occurred very early in the interval between tram and trequired- These two limiting cases, then, already reveal a key con- clusion of our analysis: the posterior distribution of A for both very large and very small values will have the shape of the prior, just scaled by different constants. Only when A is neither very large nor very small - or, more precisely, when A(tamorgo - Item) Ad 1 - do the data and the prior both inform the posterior probability at a roughly equal level. The Bayes Factor. In this context, note that the probabil- ity in equation (3) depends crucially on two time differences, At, E ! emerge — train and At2 E Inquired — tram, and that the ratio of the likelihood function at large A to its value at small A goes roughly as PldatallargeA] Ate R P[datalsmallA] At, [5] A t 104 I le 104 -3 10' lo° 10 ' io' Optimistic 1 OS 0.6 10.7 1 0.6 0.5 0.0 0.3 0.2 0.1 0 -3 ••• thelorre LOD I-3) twunil (-31 Posterior: Sold PrIonDathed -2 „ mai,pa (tin syr, 2 3 -2 Ix 2 Fig. 1. PDF and CDF of A for uniform. logarithmic, and inverse- uniform priors, for model Optimistic, with Amin = 10-3Cyr-1 and Amax = 1030yr-1. Top: The clashed and solid curves repre- sent. respectively, the prior and posterior probability distribution functions (PDFs) of A under three different assumptions about the nature of the prior. The green curves are for a prior that is uniform on the range OGyr CAS Amax ("Uniform"); the blue are for a prior that is uniform in the log of A on the range —3 ≤ log A < 3 ("Log (-3)"); and the red are for a prior that is uniform in A-1 on the interval 10-3Cyr < A-1 < 103Gyr ("InvUnif (-3)"). Bottom: The curves represent the cumulative distribution functions (CDFs) of A. The ordinate on each curve represents the integrated probabil- ity front 0 to the abscissa (color and line-style schemes are the same as in the top panel). For a uniform prior. the posterior CDF traces the prior almost exactly. In this case, the posterior judgment that A is probably large simply reflects the prior judgment of the dis- tribution of A. For the prior that is uniform in A-1 (InvUnif), the posterior judgment is quite opposite - namely, that A is probably quite small - but this judgment is also foretold by the prior, which is traced nearly exactly by the posterior. Fbr the logarithmic prior, the datum (that life on Earth arose within a certain time window) does influence the posterior assessment of A. shifting it in the di- rection of making greater %slues of A more probable. Nevertheless, the posterior probability is -.42% that A < 1Gyr -1. Lower Amm and/or lower Amax would further increase the posterior probability of very low A, for any of the priors. R is called the Bayes factor or Bayes ratio and is sometimes employed for model selection purposes. In one conventional interpretation [23], R < 10 implies no strong reason in the the rules of conditional probabity. P(RIR. PIE. RIM)/PIRIM). Slaw E entails R, the numerate, on the rrd,Yhanel side is sing* equal to P(.51M). volich means that the pewious equation reduces to equation (3). 31241 advances the darn based on theoretical armaments that me eriticalhr reevaluated in (25) Spiegel & Turner PNAS I Issue Date I Volume I Issue Number 1 3 EFTA01071748 data alone to prefer the model in the numerator over the one in the denominator. For the problem at hand, this means that the datum does not justify preference for a large value of A over an arbitrarily small one unless equation (5) gives a result larger than roughly ten. Since the likelihood function contains all of the informa- tion in the data and since the Bayes factor has the limiting behavior given in equation 5, our analysis in principle need not consider priors. If a small value of A is to be decisively ruled out by the data, the value of R must be much larger than unity. It is not for plausible choices of the parameters (see Table l), and thus arbitrarily small values of A can only be excluded by some adopted prior on its values. Still, for illustrative purposes: we now proceed to demonstrate the in- fluence of various possible A priors on the A posterior. The Prior Term. To compute the desired posterior probability, what remains to be specified is Pprior[M]: the prior joint prob- ability density function (PDF) of A, tmin, /max, and SL,voivo. One approach to choosing appropriate priors for /min, tmax. and dt„,:„No, would be to try to distill geophysical and pale- °biological evidence along with theories for the evolution of intelligence and the origin of life into quantitative distribution functions that accurately represent prior information and be- liefs about these parameters. Then, in order to ultimately calculate a posterior distribution of A, one would marginalize over these "nuisance parameters." However, since our goal is to evaluate the influence of life's early emergence on our posterior judgment of A (and not of the other parameters), we instead adopt a different approach. Rather than calculat- ing a posterior over this 4-dimensional parameter space, we investigate the way these three time parameters affect our in- ferences regarding A by simply taking their priors to be delta functions at several theoretically interesting values: a purely hypothetical situation in which life arose extremely quickly, a most conservative situation, and an in between case that is also optimistic but for which there does exist some evidence (see Table 1). For the values in Table 1, the likelihood ratio R varies from to 300. with the parameters of the "optimistic" model giving a borderline significance value of R = 15. Thus, only the hypothetical case gives a decisive preference for large A by the Bayes factor metric: and we emphasize that there is no direct evidence that abiogenesis on Earth occurred that early, only 10 million years after conditions first permitted it!3 We also lack a first-principles theory or other solid prior information for A. We therefore take three different functional forms for the prior — uniform in A, uniform in A-1 (equivalent to saying that the mean tune until life appears is uniformly distributed). and uniform in logic, A. For the uniform in A prior, we take our prior confidence in A to be uniformly dis- tributed on the interval 0 to Am„„ = 1000 Cyr-1 (and to be 0 otherwise). For the uniform in A-1 and the uniform in logio[A] priors, we take the prior density functions for A-1 and log10(A], respectively, to be uniform on Amu, < A ≤ Am„„ (and 0 otherwise). For illustrative purposes, we take three values of Amu,: 10-22Cyr-1, 10-11Cyr-1, and 10-3Cyr-1: corresponding roughly to life occuring once in the observable Universe, once in our galaxy, and once per 200 stars (assuming one Earth-like planet per star). In standard Bayesian terminology, both the uniform in A and the uniform in A-1 priors are said to be highly "informa- tive." This means that they strongly favor large and small, respectively, values of A in advance, i.e., on some basis other than the empirical evidence represented by the likelihood term. For example, the uniform in A prior asserts that we know on some other basis (other than the early emergence of life on Earth) that it is a hundred times less likely that A is less than 10-3Gyr" than that it is less than 0.1Cyr-1. The uniform in A-1 prior has the equivalent sort of preference for small A values. By contrast, the logarithmic prior is relatively "unin- formative" in standard Bayesian terminology and is equivalent to asserting that we have no prior information that informs us of even the order-of-magnitude of A. In our opinion, the logarithmic prior is the most appropri- ate one given our current lack of knowledge of the process(es) of abiogenesis, as it represents scale-invariant ignorance of the value of A. It is, nevertheless, instructive to carry all three pri- ors through the calculation of the posterior distribution of A, because they vividly illuminate the extent to which the result depends on the data vs the assumed prior. Comparison with Previous Analysis. Using a binomial proba- bility analysis, Lineweaver St Davis [9] attempted to quantify q, the probability that life would arise within the first billion years on an Earth-like planet. Although the binomial distri- bution typically applies to discrete situations (in contrast to the continuous passage of time, during which life might arise), there is a simple correspondence between their analysis and the Poisson model described above. The probability that life would arise at least once within a billion years (what [9] call q) is a simple transformation of A, obtained from equation (2), with Ati = 1 Cyr: q = _ c(A)(1Gyr) or A = Intl — 91/(1Cyr). [6] In the limit of A(1Gyr) c 1, equation (6) implies that q is equal to A(IGyr). Though not cast in Bayesian terms, the analysis in [9] draws a Bayesian conclusion and therefore is based on an implicit prior that is uniform in q. As a result, it is equivalent to our uniform-A prior for small values of A (or q), and it is this implicit prior, not the early emergence of life on Earth, that dominates their conclusions. The Posterior Probability of Abiogenesis We compute the normalized product of the probability of the data given A (equation 3) with each of the three priors (uni- form, logarithmic, and inverse uniform). This gives us the Bayesian posterior PDF of A, which we also derive for each model in Table 1. Then, by integrating each PDF from —oo to A, we obtain the corresponding cumulative distribution func- tion (CDF). Figure 1 displays the results by plotting the prior and posterior probability of A. The top panel presents the PDF, and the bottom panel the CDF, for uniform, logarithmic, and inverse-uniform priors, for model Optimistic, which sets At, (the maximum time it might have taken life to emerge once Earth became habitable) to 0.2 Cyr, and At3 (the time life had available to emerge in order that intelligent creatures would have a chance to evolve) to 3.0 Cyr. The clashed and solid curves represent, respectively, prior and posterior probability functions. In this figure, the priors on A have Amin = 10-3Cyr-1 and Amax = 103Gyr-1. The green, blue, and red curves are calculated for uniform, logarithmic, and inverse-uniform priors, respectively. The results of the corre- sponding calculations for the other models and bounds on the assumed priors are presented in the Supporting Information, but the cases shown in Fig. 1 suffice to demonstrate all of the important qualitative behaviors of the posterior. In the plot of differential probability (PDF; top panel), it appears that the inferred posterior probabilities of different values of A are conditioned similarly by the data (leading to 4 I inwri.pnas.ordegi/doi/10.1073/pnas.0709640104 Spiegel & Turner EFTA01071749 0.9 E 0.7 0.8 p 03 13 0.3 0.1 0 -3 Independent Lite. log. prior • • „ • pooetor eanz1(011141101) • -1 0 1 2 14111,01M G14-1) Fig. 2. CDF of A. fee abiogenesis with independent lineage, for bgarithmit prior. Arm. = 10- aGyr -1, Amex = 103C;yr-1. A discovery that life arose inde. pendently on Mars and Earth or on an exoplanet and Earth -or that it arose a second, independent, time on Earth - would significantly reduce the posterior probability of kw A. 2 a -16 -le -20 -22 Medan Vata of ). 1-e Lowereetntl 2-a Lows Ekemcl -26 -50 -75 Optimistic 0.0...• 101 Cyril 1-100 -80 -60 -40 -20 0 -20 -t8 -18 -14 -12 -ID -6 -6 -4 -2 ix mon') Fig. 3. Loom bound en A for logarithmic prier, Hypothetical model. The three curves depict median (50%). 1.47 (68.3%). and 247 (95.4%) lows bounds on A, as a function of Amin. a jump in the posterior PDF of roughly an order of magni- tude in the vicinity of A •• 0.5 Cyr-1). The plot of cumulative probability, however, immediately shows that the uniform and the inverse priors produce posterior CDFs that are completely insensitive to the data. Namely, small values of A are strongly excluded in the uniform in A prior case and large values are equally strongly excluded by the uniform in A-1 prior, but these strong conclusions are not a consequence of the data, only of the assumed prior. This point is particularly salient, given that a Bayesian interpretation of [9] indicates an im- plicit uniform prior. In other words, their conclusion that q cannot be too small and thus that life should not be too rare in the Universe is not a consequence of the evidence of the early emergence of life on the Earth but almost only of their particular parameterization of the problem. For the Optimistic parameters, the posterior CDF com- puted with the uninformative logarithmic prior does reflect the influence of the data, making greater values of A more probable in accordance with one's intuitive expectations. However, with this relatively uninformative prior, there is a significant probability that A is very small (12% chance that A < !Cyr-1). Moreover, if we adopted smaller Anen, smaller Amax, and/or a larger Ati/A/2 ratio, the posterior probability of an arbitrarily low A value can be made acceptably high (see Fig. 3 and the Supporting Information). Independent Abiogenesis. We have no strong evidence that life ever arose on Mars (although no strong evidence to the contrary either). Recent observations have tenatively sug- gested the presence of methane at the level of ••20 parts per billion (ppb) [26], which could potentially be indicative of bi- ological activity. The case is not entirely clear, however, as alternative analysis of the same data suggests that an upper limit to the methane abundance is in the vicinity of •••3 ppb [27]. If, in the future, researchers find compelling evidence that Mars or an exoplanet hosts life that arose independently of life on Earth (or that life arose on Earth a second, inde- pendent time [28, 29]), how would this affect the posterior probability density of A (assuming that the same A holds for both instances of abiogenesis)? If Mars, for instance, and Earth share a single A and life arose arise on Mars, then the likelihood of Mars' A is the joint probability of our data on Earth and of life arising on Mars. Assuming no panspermia in either direction, these events are independent: 'VIM] = (1 — exPE—A(tietse — ert s)]) 1 - exp[-A(trt d - tEr)] [7] For Mars, we take /MY: = rein; = Gyr and 417:" = 0.5 Gyr. The posterior cumulative probability distribution of A, given a logarithmic prior between 0.001 Gyr-1 and 1000 Gyr-1, is as represented in Fig. 2 for the case of find- ing a second, independent sample of life and, for compari- son, the Optimistic case for Earth. Should future researchers find that life arose independently on Mars (or elsewhere), this would dramatically reduce the posterior probability of very low A relative to our current inferences. I — exp[—A(t,Pg„thie _ hearth)] Arbitrarily Low Posterior Probability of A. We do not actu- ally know what the appropriate lower (or upper) bounds on A are. Figure 3 portrays the influence of changing Arnim on the median posterior estimate of A, and on 1-a and 2-a confi- dence lower bounds on posterior estimates of A. Although the median estimate is relatvely insensitive to Amin, a 2-a lower bound on A becomes arbitrarily low as Amin decreases. Conclusions Within a few hundred million years, and perhaps far more quickly, of the time that Earth became a hospitable location for life, it transitioned from being merely habitable to being inhabited. Recent rapid progress in exoplanet science sug- gests that habitable worlds might be extremely common in our galaxy [30, 31, 32, 33], which invites the question of how often life arises, given habitable conditions. Although this question ultimately must be answered empirically, via searches for biomarkers [34] or for signs of extraterrestrial technology (35], the early emergence of life on Earth gives us some infor- mation about the probability that abiogenesis will result from early-Earth-like conditions. A Bayesian approach to estimating the probability of abio- genesis clarifies the relative influence of data and of our prior 41.4b note that the ownparatively very fate emergence of rate techngogy on Earth could. anal- *gaudy. be takn as an Ai:hear:on (alant a weak one because of our single datum) that radio technology m ee be mire in our colaaY Spiegel & Tana PNAS I Issue Date I Volume I Issue Number 1 5 EFTA01071750 beliefs. Although a "best guess" of the probability of abio- genesis suggests that life should be common in the Galaxy if early-Earth-like conditions are, still, the data are consis- tent (under plausible priors) with life being extremely rare, as shown in Figure 3. Thus, a Bayesian enthusiast of extrater- restrial life should be significantly encouraged by the rapid appearance of life on the early Earth but cannot be highly confident on that basis. Our conclusion that the early emergence of life on Earth is consistent with life being very rare in the Universe for plau- sible priors is robust against two of the more fundamental simplifications in our formal analysis. First, we have assumed that there is a single value of A that applies to all Earth-like planets (without specifying exactly what we mean by "Earth- like"). If A actually varies from planet to planet, as seems far snore plausible, anthropic-like considerations imply plan- ets with particularly large A values will have a greater chance of producing (intelligent) life and of life appearing relatively rapidly, i.e., of the circumstances in which we find ourselves. Thus, the information we derive about A from the existence and early appearance of life on Earth will tend to be biased towards large values and may not be representative of the value of A for, say, an "average" terrestrial planet orbiting within the habitable zone of a main sequence star. Second, our formulation of the problem analyzed in this paper im- plicitly assumes that there is no increase in the probability of intelligent life appearing once has elapsed following the abiogenesis event on a planet. A more reasonable model in which this probability continues to increase as additional time passes would have the same qualitative effect on the cal- culation as increasing dtevowo. In other words. it would make I. Chyba. C. F. & Hand. K. P. ASTROBIOLOGY: The Study of the Living Universe. Ann. Rev. Au,. Ap. 43. 31-74 (2005). 2. Des Maras. D. & Walter. M. Au/obit:dopy: exploring the origins. evolution, and dis- tribution of Ilk in the universe. Annual review of ecology and systematks 397-420 (1999). 3. Dee Marais. D. J. et al. Remote Sensing of Planetary ProPetits and Biotignstures on Extremist Tenettriel Planet,. AstrolOology 2. 153-181 (2002). 4. Seater. 5.. Tome. E. L.. Schafer. J. & Ford. E. 8. Vegetation's Red Edge: A Pos- sible Spectroscopic Illosignature of Extraterrestrial Plants. Astrobiology S. 372-390 (2005). .rts• utre-staceonoo. S. Kasting. J. F.. INhitnre. O. P. & Reynolds. R. T. Habitable Zones around Main Sequence Stan. Icarus 101. 108-128 (1993). 6. Sao,. F. et al. Habitable planets around the star Grin StIl? Alone. Astrophys. 476. 1373-1387 (2007). ant. ano 7. Spiegel. 0. 5.. Meeou. K. & Scharf. C. A. Habitable Climates. Astrophys. J. 681. 1609-1623 (2008). ant. O11 5816. B. Carter. B. The Anthropk Principle and its Implications for Biological Evolution. Royal Society of London Philcoophical Transactions Series A 310. 147-363 (1983). 9. Lbseweaver. C. H. & Davin. T. M. Does the Rapid Appearance of Life on Earth Surest that Life I, Common in the Universe" Astrobialogy 2. 293-304 (2002). essv:nev•M/030:01.1. 10. Brewer. B. J. The Implication, of the Early Formation of Life on Earth. Argiv elobits (2008). 41:07 41.a. 11. Korptb. E. J. Statistics of One: What Earth Can and Can't Tell us About Life in the Universe. v1)00:1108.0003 (2011). lice con. 12. van Zullen. M. A.. Lapland. A. & Arrhenius. C. Reassessing the evidence for the earliest traces of life. Nat. 418.627-630 (2002). 13. Westall. F. Early life on earth: the ancient fossil record. Agrobiology; Future Per- spectives 287-316 (2005). 14. MOOrtath. S. Oldest rocks. earliest life. heaviest impacts. and the hadea arChaean transition. Applied Geochemistry 20. 819-824 (2005). 15 Norman. A. & Friend. C. Petrography and geochemistry of apatkes in banded son formation. along,. w. greenland: Consequences for oldest life evidence Precambrian Research IS?. 100-106 (2006). 16 Buick. R The earliest records of life on Earth. In St ran. W. T. & Baron. J. A. fed,.) Planets and Me: the emerging science of astrobiology (Cambridge University Press. 2007). 17. Sullivan. W. & Baron. J. Planets and life: the emerging science of astrobiology (Cambridge University Port 2007). the resulting posterior distribution of A even less sensitive to the data and more highly dependent on the prior because it would make our presence on Earth a selection bias favoring planets on which abiogenesis occurred quickly. We had to find ourselves on a planet that has life on it, but we (lid not have to find ourselves (1) in a galaxy that has life on a planet besides Earth nor (ii) on a planet on which life arose multiple, independent times. Learning that either (i) or (ii) describes our world would constitute data that are not subject to the selection effect described above. In short, if we should find evidence of life that arose wholly idependently of us - either via astronomical searches that reveal life on an- other planet or via geological and biological studies that find evidence of life on Earth with a different origin from us - we would have considerably stronger grounds to conclude that life is probably common in our galaxy. With this in mind, research in the fields of astrobiology and origin of life stud- ies might, in the near future, help us to significantly refine our estimate of the probability (per unit time, per Earth-like planet) of abiogenesis. ACKNOWLEDGMENTS. We thank Adam Brown, Adam Burrows. Chris Chyba, Scott Gaudi, Aaron Goldman, Alan Guth, Larry Guth, Laura Landweber. Tullis On• stet. Caleb Scharf, Stanley Spiegel, Josh Winn, and Neil Zimmerman for thoughtful discussions. ELT is grateful to Cad Boettiger (or calling this problem to his atten• lion some years ago. DSS acknowledges support from NASA grant NNX07AG80G, from JPL/Spitzer Agreements 1328092, 1348668, and 1312647, and gratefully ac• knew/ledges support from NSF grant AST-0807444 and the Keck Fellowship. ELT gratefully acknowledge support from the World Premier International Researds Cen• to Initiative (Kavli•IPMU), MEXT. Japan and the Research Center (or the Early Universe at the University of Tokyo as well as the hospitality of its Department of Physics. Finally, we thank two anonymous referee for comments that materially improved this manuscript. Sugitani. K. et al. 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Glieee 581d is the First Discovered T./nobleman Emplane' in the Habitable Zone. Astrophys. J. Let. 733. 148+ (2011). 1106 34. Kahenagget. L & &alas. F. Blomanters set in contort. AtXlv e.pintS 0710.0881 (2007). eao) 31. Tartet.1 Th Search for Extraterrestrial Intelligence (SETT). Ann. Rev. Astor. Ap. 39. 511-548 (2001). 6 I www.pnas.agicgi/doi/10.1073/pnas.0709640104 Spiegel & Turner EFTA01071751 Spegel & Turner PNAS I Issue Data I Volume I !nue Number f 7 EFTA01071752 Supplementary Material Formal Derivation of the Posterior Probability of Abiogenesis Let to be the time of abiogenesis and t, be the time of the emergence of intelligence (to, /emerge, and enquired are as defined in the text: to is the current age of the Earth; iornerge is the upper limit on the age of the Earth when life first arose; and ttcoui„ d is the maximum age the Earth could have had when life arose in order for it to be possible for sentient beings to later arise by to). By "intelligence", we mean organisms that think about abiogenesis. Furthermore, let E = (min < t o < (emerge = < to < ( required I = /min < t, ≤ to ,A4 = "The Poisson rate parameter has value A" We assert (perhaps somewhat unreasonably) that the probability of intelligence arising (1) is independent of the actual time of abiogeneis (G), so long as life shows up within ti.ol„o„d (ft). P[IIR,M, t0I = PVIR. MI [8] Although the probability of intelligence arising could very well be greater if abiogenesis occurs earlier on a world, the conse- quence of relaxing this assertion (discussed in the Conclusion and elsewhere in the text) is to increase the posterior probability of arbitrarily low A. Using the conditional version of Bayes's theorem, PEW ft, Mal P ft, MI PI/ IR, A4, GI PIG IR, MI [9] and Eq. (8) then implies that P[GIR, A4,1] = P[taIR, M]. An immediate result of this is that NEP, A4, /1 = Piga, .A4] . [10] We now apply Bayes's theorem again to get the posterior probability of A, given our circumstances and our observations: /] x P[MIR, /I P[MIE, — [11] P[E,R, Note that, since to < tr, E And, as discussed in the main text, P[E1R, MI = P[EIMI/P[RIM]. Finally, since we had to find ourselves on a planet on which R and / hold, these conditions tell us nothing about the value of A. In other words, P[MIR, = P[M]. We therefore use Eq. (10) to rewrite Eq. (11) as the posterior probability implied in the text: P x P(A4] P[MIE, f] [12] P[E, f] Model-Dependence of Posterior Probability of Abiogenesis In the main text, we demonstrated the strong dependence of the posterior probability of life on the form of the prior for A. Here, we present a suite of additional calculations, for different bounds to A and for different values of At, and At2. Figure 4 displays the results of analogous calculations to those of Fig. 1, for three sets model of parameters (Hypothetical., Optimistic. Conservative) and for three values of Amin (10-22Gyr-1, 10- "Gyr 10-3Gyr-r). For all three models, the posterior CDFs for the uniform and the inverse-uniform priors almost exactly match the prior CDFs, and, hence, are almost completely insensitive to the data. For the Conservative model (in which Alt = 0.8 Cyr and At2 = 0.9 Gyr - certainly not ruled out by available data), even the logarithmic prior's CDF is barely sensitive to the observation that there is life on Earth. Finally, the effect of Ste,,,,. - the minimum timescale required for sentience to evolve - is to impose a selection effect that becomes progressively more severe as otayolye approaches to — (emerge. Figure 5 makes this point vividly. For the Optimistic model, posterior probabilities are shown as color maps as functions of A (abscissa) and &evolve (ordinate). At each horizontal cut across the PDF plots (left column), the values integrate to unity, as expected for a proper probability density function. For short values of oto„oh.„, the selection effect (that intelligent creatures take some time to evolve) is unimportant, and the data might be somewhat informative about the true distribution of A. For larger values of 6tovoivo, the selection effect becomes more important, to the point that the probability of the data given A approaches I, and the posterior probability approaches the prior. 8 I vnwe.pnas.terfegi/doi/10.1073/pnas.0709640104 Spiegel & Turner EFTA01071753 1040 IT" . . — es . II . . ..., Unitan LO3 -1I1 • -221 • 11-3) Irr.Vrtl C-11) • •,.. vy.tall (-22) • • • • Posisdat Said • ° Prix:Dashed • • .,. • • • • • • • • Hypothetical • •,. . . -22 -19 -16 -13 - 1 -a -6 109,471 P. hale) to° te io 104o 104) -22 • ftelefet SOW • Mon Dashed • • • • • • • • Optimistic -3 0 •-• • 3 to° 10-s 10-t -19 -IS -13 - 1 4 -S , 103,0IM 1/ 1• 3)T- 1 0 3 .1 Vattern Lea at t.22, Inr1J0.1 it) InAfrol -22) Poston= Sold Srke:Osiliel Conservative -19 -16 -13 -II 4 -5 3 0,1,) InGYr.) -3 0 09 Os 307 1 OA OS 01 0 03 02 01 0 -22 -IS -IS -13 -11 4 -S -3 0 -22 -19 -IS -13 -IT -6 -5 -3 0 3 le34311/ 1,0K ,1 100412 4M02(1) -19 -16 -13 -It 4 -5 109•P inOKI) 4 0 3 Fig. 4. PDF (left) and CDF (right) of A (or uniform. logarithmic, and inverse-uniform priors, for models Hypothetical (top), Optimistic (middle). Conservative (bottom). Curves are shown for prices xith Amin = I0-22Cyr —I, Amin = 10— Gr .— i s and Amin = 10 —3Gyr —I. The uniform and inverse-uniform priors lead to CDFs that are completely insensitive to the data (or all three models; (or the Conservative model, even the logarithmic prior is insensitive to the data. Spiegel & Turner PNAS I Issue Date I Volume I lout Number I 9 EFTA01071754 0. 2 -3 -2 -1 leg,,131 11. inewr-5 3. 0. 2 -3 -2 -1 0 100,104 IX new') 3. 0. 3 2 $7. 9- E Y 3 2 `t- IS 3 2 03 -3 0 2 3 loo,p1 Ain Gra) 3.5 2 5 2 ; 4 1- 5 -a -3 -2 -1 baioN 0. 0..GyrI) I -3 2 3 -1 Fig. 5. The influence of Stovok.e. Uniform prior (top). logarithmic prior (middle). and inversewnifonn prier (bottom). l0 in GYr k 51 PDF (left) and CDF (right). Aside Iran Stood", parameters are set to the Optimistic model with Amin = 10-3Gyr —1 in the logarithmic and inverse-uniform cases. J 0: 2.5 0 9 Ib ;4 -1; 3 2 S 2 a SE A /4 3 2 2 3 10 I www.pnasordegi/doi/10.1073/pnas.0709640104 Spiegel & Turner EFTA01071755