EFTA01071746.pdf

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 00540.1Dept. 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 Life arose on Earth sometime in the first few hundred million years Any inferences about the probability of life arising (given after the young planet had cooled to the point that it could support the conditions present on the early Earth) must be informed water-based organisms on its surface. The early emergence of life by how long it took for the first living creatures to evolve. By on Earth has been taken as evidence that the probability of abiogen- definition, improbable events generally happen infrequently. arXiv:1107.3835v4 [astro-ph.EP] 13 Apr 2012 esis is high. if starting from young-Earth-like conditions. We revisit It follows that the duration between events provides a metric this argument quantitatively in a Bayesian statistical framework. By constructing a simple model of the probability of abiogenesis. we (however imperfect) of the probability or rate of the events. calculate a Bayesian estimate of its posterior probability, given the The time-span between when Earth achieved pre-biotic condi- data that life emerged fairly early in Earth's history and that. billions tions suitable for abiogenesis plus generally habitable climatic of years later. curious creatures noted this fact and considered its conditions Is, 6, 7] and when life first arose, therefore, seems implications. We find that. given only this very limited empirical to serve as a basis for estimating A. Revisiting and quantifying information, the choice of Bayesian prior for the abiogenesis proba- this analysis is the subject of this paper. bility parameter has a dominant influence on the computed posterior We note several previous quantitative attempts to address probability. Although terrestrial life's early emergence provides evi- this issue in the literature, of which one [8] found, as we dence that life might be common in the Universe if early-Earth-like do, that early abiogenesis is consistent with life being rare, conditions are. the evidence is inconclusive and indeed is consistent with an arbitrarily low intrinsic probability of abiogenesis for plausible and the other [9] found that Earth's early abiogenesis points uninformative priors. Finding a single case of life arising indepen- strongly to life being common on Earth-like planets (we com- dently of our lineage (on Earth. elsewhere in the Solar System. or pare our approach to the problem to that of [9] below, in- on an extrasolar planet) would provide much stronger evidence that cluding our significantly different results).' lAirthermore, an abiogenesis is not extremely rare in the Universe. argument of this general sort has been widely used in a qual- itative and even intuitive way to conclude that A is unlikely Astrobiology to be extremely small because it would then be surprising for abiogenesis to have occurred as quickly as it did on Earth Abbreviations: Gr. gigayear (10" years); PDF, probability density function; CDF, [12, 13, I4, 15, 16, 17, 18]. Indeed, the early emergence of life cumulative distribution function 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 Introduction motivation of this paper is to determine the quantitative va- Astrobiology is fundamentally concerned with whether ex- lidity of this inference. We emphasize that our goal is not to traterrestrial life exists and, if so, how abundant it is in the derive an optimum estimate of A based on all of the many lines of available evidence, but simply to evaluate the implication Universe. The most direct and promising approach to answer- ing these questions is surely empirical, the search for life on of life's early emergence on Earth for the value of A. other bodies in the Solar System [1, 2] and beyond in other planetary systems [3, 4]. Nevertheless, a theoretical approach A Bayesian Formulation of the Calculation is possible in principle and could provide a useful complement to the more direct lines of investigation. Bayes's theorem [21] can be wrii Ion as P[MID] = In particular, if we knew the probability per unit time (11DIMNPpno,.[MD/P[D]. Here. we take M to be a model and per unit volume of abiogenesis in a pre-biotic environ- and V to be data. In order to us• this equation to evalu- ment as a function of its physical and chemical conditions ate the posterior probability of abiogenesis, we must specify and if we could determine or estimate the prevalence of such appropriate M and D. 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. t daverkisedu There does, however, appear to be one possible way of fi- Reserved for Publication Footnotes 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." 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 Models of to = 4.5 Cyr-Old Planets simplistic, model is that it is a Poisson process during a pe- o e thetical Conserv.' Consery .2 tialsci.c riod of time from foam until tmax. In this model, the conditions on a young planet preclude the development of life for a time tnti„ 0.5 0.5 0.5 0.5 period of I rmo after its formation. Furthermore, if the planet temorge 0.51 1.3 1.3 0.7 remains lifeless until t„,„ has elapsed, it will remain lifeless thereafter as well because conditions no longer permit life to tmax 10 1.4 10 10 arise. For a planet around a solar-type star, tnax is almost Sterols. 1 2 3.1 1 certainly 10 Gyr (10 billion years, the main sequence life- trequired 3.5 1.4 1.4 3.5 time of the Sun) and could easily be a substantially shorter period of time if there is something about the conditions on All 0.01 0.80 0.80 0.20 a young planet that are necessary for abiogenesis. Between a:2 3.00 0.90 0.90 3.00 these limiting times, we posit that there is a certain probabil- ity per unit time (A) of life developing. For train < t < Luau: 300 1.1 1.1 15 then, the probability of life arising n times in time t is All times are in Cyr. Two "Conservative" (Conserv.) models are shown, to indicate that !required may be limited either by a small P[A, n, = Ppobt. =C AO -I mm) {AO !min)]" value of /max ("Conserv. i"), or by a large value of ote„3,.. ("Conserv.2"). [1 ] where t is the time since the formation of the planet. This formulation could well be questioned on a number of bly high (in our language, A is probably large). This stan- grounds. Perhaps most fundamentally, it treats abiogenesis dard argument neglects a potentially important selection ef- as though it were a single instantaneous event and implicitly fect, namely: On Earth, it took nearly 4 Gyr for evolution to assumes that it can occur in only a single way (i.e., by only a lead to organisms capable of pondering the probability of life single process or mechanism) and only in one type of physical elsewhere in the Universe. If this is a necessary duration, then environment. It is, of course, far more plausible that abiogen- it would be impossible for us to find ourselves on, for example, esis is actually the result of a complex chain of events that a (-4.5-Gyr old) planet on which life first arose only after the take place over some substantial period of time and perhaps passage of 3.5 billion years (221. On such planets there would via different pathways and in different environments. How- not yet have been enough time for creatures capable of such ever, knowledge of the actual origin of life on Earth, to say contemplations to evolve. In other words, if evolution requires nothing of other possible ways in which it might originate, is 3.5 Gyr for life to evolve from the simplest forms to intelligent, so limited that a more complex model is not yet justified. In questioning beings, then we had to find ourselves on a planet essence, the simple Poisson event model used in this paper where life arose relatively early, regardless of the value of A. attempts to "integrate out" all such details and treat abio- In order to introduce this constraint into the calculation genesis as a "black box" process: certain chemical and phys- we define 8t„.„4, as the minimum amount of time required af- ical conditions as input produce a certain probability of life ter the emergence of life for cosmologically curious creatures emerging as an output. Another Sue is that A, the probabil- to evolve, tom,ngo as the age of the Earth from when the earliest ity per unit time, could itself be a function of time. In fact, extant evidence of life remains (though life might have actu- the claim that life could not have arisen outside the window ally emerged earlier), and to as the current age of the Earth. (tni„,tro.) is tantamount to saying that A = 0 for t ≤ !min The data, then, are that life arose on Earth at least once, ap- and for t ≥ tmax. Instead of switching from 0 to a fixed value proximately 3.8 billion years ago, and that this emergence was instantaneously, A could exhibit a complicated variation with early enough that human beings had the opportunity subse- time. If so, however, P[A,n,fi is not represented by the Pois- quently to evolve and to wonder about their origins and the son distribution and eq. (1) is not valid. Unless a particular possibility of life elsewhere in the Universe. In equation form, (non top-hat-function) tune-variation of A is suggested on the- !emerge < to — Otevolvo• oretical grounds, it seems unwise to add such unconstrained complexity. The Likelihood Term. We now seek to evaluate the P[DIM] A further criticism is that A could be a function of n: it term in Bayes's theorem. Let ta,,,oi„d a min[to — could be that life arising once (or more) changes the probabil- &evoke, Gnash Our existence on Earth requires that life ap- ity per unit time of life arising again. Since we are primarily peared within ! required. In other words, t„„,„fr od is the max- interested in the probability of life arising at all - i.e., the imum age that the Earth could have had at the origin of probability of n 0 0 - we can define A simply to be the value life in order for humanity to have a chance of showing up appropriate for a prebiotic planet (whatever that value may by the present. We define Se to be the set of all Earth-like be) and remain agnostic as to whether it differs for n ≥ 1. worlds of age approximately to in a large, unbiased volume Thus, within the adopted model, the probability of life aris- and L[1] to be the subset of St on which life has emerged ing is one minus the probability of it not arising: within a time t. Litrecperoal is the set of planets on which life emerged early enough that creatures curious about abio- Plife = I — PPotsson EA, 0,t] = 1 n• C —Mt . [2] genesis could have evolved before the present (to), and, pre- suming te=„ = < tpmo,„d (which we know was the case for A Minimum Evolutionary Time Constraint. Naively, the single Earth), glemargo] is the subset of Wrequiradl on which life datum informing our calculation of the posterior of A appears emerged as quickly as it did on Earth. Correspondingly, Nst, to be simply that life arose on Earth at least once, approxi- NG, and NL, are the respective numbers of planets in sets mately 3.8 billion years ago (give or take a few hundred million Se, L[trequirea], and gtomorgel. The fractions sot, a Mr /Nse 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- 2 An °hematite nuy to derive equation (3) is to let E = "abiocenSs occurred between emi r, and ined for inhabitants, the probability of abiogenesis is proba- 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- 10' tion on which life emerged within t,„,„,o. The ratio r a Optimistic lo° 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- 10 ' selves on such a planet in the set Ltr in order to write and read about this topic, the ratio r characterizes the probability io' of the data given the model if the probability of intelligent A observers arising is independent of the time of abiogenesis 104 ••• t (so long as abiogenesis occurs before tre„,„i„d). (This last as- I le thelorre sumption might seem strange or unwarranted, but the effect LOD I-3) of relaxing this assumption is to make it more likely that we twunil (-31 would find ourselves on a planet with early abiogenesis and Posterior: Sold therefore to reduce our limited ability to infer anything about PrIonDathed A from our observations.) Since co‘ = I— PpoinorP, 0, immerge] and <At =1 - Proh,..4A,O,t,„,qui,,il, we may write that 104 -3 -2 „ 2 3 mai,pa (tin syr, p[, I A4] I onterge Imln)] 1 [3] 1— exp[ exp[—A(4.„,vd„d — /min)] OS if , min < Leman,. < ta..quired (and P[DI.A4] = 0 otherwise). This 0.6 is called the "likelihood function," and represents the proba- 10.7 bility of the observation(s), given a particular model. It is 1 0.6 via this function that the data "condition" our prior beliefs about A in standard Bayesian terminology. 0.5 Limiting Behavior of the Likelihood. It is instructive to con- 0.0 sider the behavior of equation (3) in some interesting limits. 0.3 Fbr — Gem) C 1. the numerator and denominator 0.2 of equation (3) each go approximately as the argument of the exponential function; therefore, in this limit, the likelihood 0.1 function is approximately constant: 0 -3 -2 2 !emerge — tram Ix PEDIAll 4 [4] ...required — train Fig. 1. PDF and CDF of A for uniform. logarithmic, and inverse- uniform priors, for model Optimistic, with Amin = 10-3Cyr-1 This result is intuitively easy to understand as follows: If A and Amax = 1030yr-1. Top: The clashed and solid curves repre- is sufficiently small, it is overwhelmingly likely that abiogene- sent. respectively, the prior and posterior probability distribution sis occurred only once in the history of the Earth, and by the functions (PDFs) of A under three different assumptions about the assumptions of our model, the one event is equally likely to oc- nature of the prior. The green curves are for a prior that is uniform cur at any time during the interval between train and troquired. on the range OGyr CAS Amax ("Uniform"); the blue are for a The chance that this will occur by t,,,, o is then just the 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 fraction of that total interval that has passed by ! mange - the the interval 10-3Cyr < A-1 < 103Gyr ("InvUnif (-3)"). Bottom: result given in equation (4). The curves represent the cumulative distribution functions (CDFs) In the other limit, when A(t0moexe - train) >, 1, the numer- of A. The ordinate on each curve represents the integrated probabil- ator and denominator of equation (3) are both approximately ity front 0 to the abscissa (color and line-style schemes are the same I. In this case, the likelihood function is also approximately as in the top panel). For a uniform prior. the posterior CDF traces constant (and equal to unity). This result is even more in- the prior almost exactly. In this case, the posterior judgment that tuitively obvious since a very large value of A implies that A is probably large simply reflects the prior judgment of the dis- abiogenesis events occur at a high rate (given suitable condi- tribution of A. For the prior that is uniform in A-1 (InvUnif), the posterior judgment is quite opposite - namely, that A is probably tions) and are thus likely to have occurred very early in the quite small - but this judgment is also foretold by the prior, which interval between tram and trequired- is traced nearly exactly by the posterior. Fbr the logarithmic prior, These two limiting cases, then, already reveal a key con- the datum (that life on Earth arose within a certain time window) clusion of our analysis: the posterior distribution of A for does influence the posterior assessment of A. shifting it in the di- both very large and very small values will have the shape of rection of making greater %slues of A more probable. Nevertheless, the prior, just scaled by different constants. Only when A is the posterior probability is -.42% that A < 1Gyr-1. Lower Amm neither very large nor very small - or, more precisely, when and/or lower Amax would further increase the posterior probability A(tamorgo - Item) Ad 1 - do the data and the prior both inform of very low A, for any of the priors. the posterior probability at a roughly equal level. R is called the Bayes factor or Bayes ratio and is sometimes The Bayes Factor. In this context, note that the probabil- employed for model selection purposes. In one conventional ity in equation (3) depends crucially on two time differences, interpretation [23], R < 10 implies no strong reason in the At, E ! emerge — train and At2 E I nquired — tram, and that the ratio of the likelihood function at large A to its value at small A goes roughly as the rules of conditional probabity. P(RIR. PIE. RIM)/PIRIM). Slaw E entails R, PldatallargeA] the numerate, on the rrd,Yhanel side is sing* equal to P(.51M). volich means that the pewious Ate equation reduces to equation (3). R [ 5] P[datalsmallA] At, 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 some other basis (other than the early emergence of life on in the denominator. For the problem at hand, this means that Earth) that it is a hundred times less likely that A is less than the datum does not justify preference for a large value of A 10-3Gyr" than that it is less than 0.1Cyr-1. The uniform over an arbitrarily small one unless equation (5) gives a result in A-1 prior has the equivalent sort of preference for small A larger than roughly ten. values. By contrast, the logarithmic prior is relatively "unin- Since the likelihood function contains all of the informa- formative" in standard Bayesian terminology and is equivalent tion in the data and since the Bayes factor has the limiting to asserting that we have no prior information that informs behavior given in equation 5, our analysis in principle need us of even the order-of-magnitude of A. not consider priors. If a small value of A is to be decisively In our opinion, the logarithmic prior is the most appropri- ruled out by the data, the value of R must be much larger ate one given our current lack of knowledge of the process(es) than unity. It is not for plausible choices of the parameters of abiogenesis, as it represents scale-invariant ignorance of the (see Table l), and thus arbitrarily small values of A can only value of A. It is, nevertheless, instructive to carry all three pri- be excluded by some adopted prior on its values. Still, for ors through the calculation of the posterior distribution of A, illustrative purposes: we now proceed to demonstrate the in- because they vividly illuminate the extent to which the result fluence of various possible A priors on the A posterior. depends on the data vs the assumed prior. Comparison with Previous Analysis. Using a binomial proba- The Prior Term. To compute the desired posterior probability, bility analysis, Lineweaver St Davis [9] attempted to quantify what remains to be specified is Pprior[M]: the prior joint prob- q, the probability that life would arise within the first billion ability density function (PDF) of A, tmin, /max, and SL,voivo. years on an Earth-like planet. Although the binomial distri- One approach to choosing appropriate priors for /min, tmax. bution typically applies to discrete situations (in contrast to and dt„,:„No, would be to try to distill geophysical and pale- the continuous passage of time, during which life might arise), °biological evidence along with theories for the evolution of there is a simple correspondence between their analysis and intelligence and the origin of life into quantitative distribution the Poisson model described above. The probability that life functions that accurately represent prior information and be- would arise at least once within a billion years (what [9] call liefs about these parameters. Then, in order to ultimately q) is a simple transformation of A, obtained from equation (2), calculate a posterior distribution of A, one would marginalize with Ati = 1 Cyr: over these "nuisance parameters." However, since our goal = _ c(A)(1Gyr) is to evaluate the influence of life's early emergence on our q or A = Intl — 91/(1Cyr). [6] posterior judgment of A (and not of the other parameters), we instead adopt a different approach. Rather than calculat- In the limit of A(1Gyr) c 1, equation (6) implies that q ing a posterior over this 4-dimensional parameter space, we is equal to A(IGyr). Though not cast in Bayesian terms, the investigate the way these three time parameters affect our in- analysis in [9] draws a Bayesian conclusion and therefore is ferences regarding A by simply taking their priors to be delta based on an implicit prior that is uniform in q. As a result, it functions at several theoretically interesting values: a purely is equivalent to our uniform-A prior for small values of A (or hypothetical situation in which life arose extremely quickly, q), and it is this implicit prior, not the early emergence of life a most conservative situation, and an in between case that is on Earth, that dominates their conclusions. also optimistic but for which there does exist some evidence (see Table 1). For the values in Table 1, the likelihood ratio R varies The Posterior Probability of Abiogenesis from to 300. with the parameters of the "optimistic" We compute the normalized product of the probability of the model giving a borderline significance value of R = 15. Thus, data given A (equation 3) with each of the three priors (uni- only the hypothetical case gives a decisive preference for large form, logarithmic, and inverse uniform). This gives us the A by the Bayes factor metric: and we emphasize that there Bayesian posterior PDF of A, which we also derive for each is no direct evidence that abiogenesis on Earth occurred that model in Table 1. Then, by integrating each PDF from —oo to early, only 10 million years after conditions first permitted it!3 A, we obtain the corresponding cumulative distribution func- We also lack a first-principles theory or other solid prior tion (CDF). information for A. We therefore take three different functional Figure 1 displays the results by plotting the prior and forms for the prior — uniform in A, uniform in A-1 (equivalent posterior probability of A. The top panel presents the PDF, to saying that the mean tune until life appears is uniformly and the bottom panel the CDF, for uniform, logarithmic, and distributed). and uniform in logic, A. For the uniform in A inverse-uniform priors, for model Optimistic, which sets At, prior, we take our prior confidence in A to be uniformly dis- (the maximum time it might have taken life to emerge once tributed on the interval 0 to Am„„ = 1000 Cyr-1 (and to Earth became habitable) to 0.2 Cyr, and At3 (the time life be 0 otherwise). For the uniform in A-1 and the uniform in had available to emerge in order that intelligent creatures logio[A] priors, we take the prior density functions for A-1 would have a chance to evolve) to 3.0 Cyr. The clashed and log10(A], respectively, to be uniform on Amu, < A ≤ Am„„ and solid curves represent, respectively, prior and posterior (and 0 otherwise). For illustrative purposes, we take three probability functions. In this figure, the priors on A have values of Amu,: 10-22Cyr-1, 10-11Cyr-1, and 10-3Cyr-1: Amin = 10-3Cyr-1 and Amax = 103Gyr-1. The green, blue, corresponding roughly to life occuring once in the observable and red curves are calculated for uniform, logarithmic, and Universe, once in our galaxy, and once per 200 stars (assuming inverse-uniform priors, respectively. The results of the corre- one Earth-like planet per star). sponding calculations for the other models and bounds on the In standard Bayesian terminology, both the uniform in A assumed priors are presented in the Supporting Information, and the uniform in A-1 priors are said to be highly "informa- but the cases shown in Fig. 1 suffice to demonstrate all of the tive." This means that they strongly favor large and small, important qualitative behaviors of the posterior. respectively, values of A in advance, i.e., on some basis other In the plot of differential probability (PDF; top panel), it than the empirical evidence represented by the likelihood term. appears that the inferred posterior probabilities of different For example, the uniform in A prior asserts that we know on values of A are conditioned similarly by the data (leading to 4 I inwri.pnas.ordegi/doi/10.1073/pnas.0709640104 Spiegel & Turner EFTA01071749  Amax, and/or a larger Ati/A/2 ratio, the posterior probability of an arbitrarily low A value can be made acceptably high (see 0.9 Independent Lite. log. prior Fig. 3 and the Supporting Information). Independent Abiogenesis. We have no strong evidence that E 0.7 life ever arose on Mars (although no strong evidence to the 0.8 contrary either). Recent observations have tenatively sug- gested the presence of methane at the level of ••20 parts per p 03 billion (ppb) [26], which could potentially be indicative of bi- ological activity. The case is not entirely clear, however, as 13 0.3 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 0.1 „ •pooetor eanz1(011141101) that Mars or an exoplanet hosts life that arose independently of life on Earth (or that life arose on Earth a second, inde- 0 -3 -1 0 1 2 pendent time [28, 29]), how would this affect the posterior 14111,01M G14-1) probability density of A (assuming that the same A holds for Fig. 2. CDF of A. fee abiogenesis with independent lineage, for bgarithmit prior. both instances of abiogenesis)? Arm. = 10- aGyr -1, Amex = 103C;yr-1. A discovery that life arose inde. If Mars, for instance, and Earth share a single A and life pendently on Mars and Earth or on an exoplanet and Earth -or that it arose a second, arose arise on Mars, then the likelihood of Mars' A is the joint independent, time on Earth - would significantly reduce the posterior probability of probability of our data on Earth and of life arising on Mars. kw A. Assuming no panspermia in either direction, these events are independent: 2 'VIM] = ( 1 — exPE— A(t e i tse — ert s)]) a I — exp[—A(t,Pg„thie _ hearth)] 1 - exp[-A(trt d - tEr)] [7] Medan Vata of ). 1-e Lowereetntl For Mars, we take /MY: = rein; = Gyr and 417:" = 2-a Lows Ekemcl 0.5 Gyr. The posterior cumulative probability distribution of A, given a logarithmic prior between 0.001 Gyr-1 and -26 1000 Gyr-1, is as represented in Fig. 2 for the case of find- -50 ing a second, independent sample of life and, for compari- son, the Optimistic case for Earth. Should future researchers -16 -75 find that life arose independently on Mars (or elsewhere), this Optimistic -le 0.0...• 101 Cyri l would dramatically reduce the posterior probability of very 1-100 -80 -60 -40 -20 0 -20 low A relative to our current inferences. -22 -20 -t8 -18 -14 -12 -ID -6 -6 -4 -2 ix mon') Arbitrarily Low Posterior Probability of A. We do not actu- Fig. 3. Loom bound en A for logarithmic prier, Hypothetical model. The ally know what the appropriate lower (or upper) bounds on three curves depict median (50%). 1.47 (68.3%). and 247 (95.4%) lows bounds on A are. Figure 3 portrays the influence of changing Arnim on A, as a function of Amin. the median posterior estimate of A, and on 1-a and 2-a confi- dence lower bounds on posterior estimates of A. Although the a jump in the posterior PDF of roughly an order of magni- median estimate is relatvely insensitive to Amin, a 2-a lower tude in the vicinity of A •• 0.5 Cyr-1). The plot of cumulative bound on A becomes arbitrarily low as Amin decreases. probability, however, immediately shows that the uniform and the inverse priors produce posterior CDFs that are completely Conclusions insensitive to the data. Namely, small values of A are strongly excluded in the uniform in A prior case and large values are Within a few hundred million years, and perhaps far more equally strongly excluded by the uniform in A-1 prior, but quickly, of the time that Earth became a hospitable location these strong conclusions are not a consequence of the data, for life, it transitioned from being merely habitable to being only of the assumed prior. This point is particularly salient, inhabited. Recent rapid progress in exoplanet science sug- given that a Bayesian interpretation of [9] indicates an im- gests that habitable worlds might be extremely common in plicit uniform prior. In other words, their conclusion that q our galaxy [30, 31, 32, 33], which invites the question of how cannot be too small and thus that life should not be too rare often life arises, given habitable conditions. Although this in the Universe is not a consequence of the evidence of the question ultimately must be answered empirically, via searches early emergence of life on the Earth but almost only of their for biomarkers [34] or for signs of extraterrestrial technology particular parameterization of the problem. (35], the early emergence of life on Earth gives us some infor- For the Optimistic parameters, the posterior CDF com- mation about the probability that abiogenesis will result from puted with the uninformative logarithmic prior does reflect early-Earth-like conditions. the influence of the data, making greater values of A more A Bayesian approach to estimating the probability of abio- probable in accordance with one's intuitive expectations. genesis clarifies the relative influence of data and of our prior However, with this relatively uninformative prior, there is a significant probability that A is very small (12% chance that 41.4b note that the ownparatively very fate emergence of rate techngogy on Earth could. anal- A < !Cyr-1). Moreover, if we adopted smaller Anen , smaller *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- the resulting posterior distribution of A even less sensitive to genesis suggests that life should be common in the Galaxy the data and more highly dependent on the prior because it if early-Earth-like conditions are, still, the data are consis-