Distribution of Success Times.

Following earlier work (13, 20⇓–22), we describe abiogenesis as a uniform rate (i.e., Poisson) process, defined by a rate parameter λL. As with ref. 13, we caution that this does not imply that abiogenesis is a truly single-step, instantaneous event. Rather, we interpret the process to be a model which “integrates out” the likely complex and multistep chemistry which culminates in life. Indeed, it may be that a variety of pathways can lead to abiogenesis, but this ensemble is grouped into a single process where any of them succeeding counts as a success for the ensemble. Further, it is not necessary to strictly define what is meant by “life” here, only that the success of this Poisson process ultimately led to the geological evidence for life and that without it said evidence would not exist.

The assumption of a uniform-rate process over some time interval can at first seem problematic when one considers the stark changes to Earth’s environment over its history. However, much of this change is due to life affecting its environment and thus is a consequence of a success. The abiogenesis rate may indeed be different after life begins, but it is also irrelevant since our model cares only about the first success.

With these points in mind, we can now write that in a time interval tL, the probability of obtaining at least one successful abiogenesis event (XL>0) will bePr(XL>0;λL,tL)=1−Pr(XL=0;λL,tL),  =1−e−λLtL.[1]The time interval between successes for a Poisson process follows an exponential distribution. This can be demonstrated by noting that Eq. 1 corresponds to the probability of obtaining at least one success over the interval of time tL and thus can be understood as the cumulative probability distribution for the achievement of at least one success by that time. Accordingly, the probability density function of the first success with respect to tL must be given by the derivative of Eq. 1 with respect to time, yieldingPr(tL|λL)=λLe−λLtL.[2]We now deviate from the approach of ref. 13 by considering a second process, labeled “I” for “intelligence,” which can proceed only once the previous process (“L”) is successful. The inverse of this process’s rate parameter, λI, describes the characteristic timescale it takes for evolution to develop from the earliest forms of life, to an “intelligent observer” which is carefully defined in Accounting for Observational Constraints. We truncate the times, such that both processes have to occur within a finite time T. Since process I can proceed only once process L has occurred, then this requires tL+tI<T. The joint distribution of times tL and tI is therefore given byPr(tL,tI|λL,λI)∝λLλIe−λLtL−λItI if tL+tI<T,0 otherwise.[3]Imposing the condition of tL+tI<T serves to truncate the joint distribution and thus the above is formally a proportionality because it is not yet normalized. After normalization, the expression becomesPr(tL,tI|λL,λI)=λLλI(λL−λI)e−λLtL−λItIλL(1−e−λIT)−λI(1−e−λLT) if tL+tI<T,0 otherwise.[4]

Accounting for Observational Constraints.

Before we discuss how Eq. 4 can be updated to include observational constraints, it is first useful to define exactly what we mean by intelligence in this work. We adopt a functional view of this term and consider that a successful event from process I is defined as some kind of transition—which occurs after abiogenesis—which is fundamentally necessary for analyses such as the one presented here to be possible. In other words, this type of analysis is possible only because process I succeeded and would be impossible if it failed.

Expounding upon this, we can consider that process I results in an observer/entity/society capable of 1) obtaining and dating geological evidence pertaining to the early emergence of life, 2) the ability to model the future climatic conditions of the world such that the habitability window can be estimated, and 3) interpreting the ramifications of this information regarding the underlying rates of abiogenesis and evolution. For the sake of brevity we refer to such outcomes as intelligent observers in what follows. Formally, these three conditions are not equivalent to a technological civilization, but we argue that it is difficult to imagine how these feats would be possible in the absence of one.

In what follows, we attribute process I to correspond to the emergence of human civilization and thus further assume that no previous Earth-dwelling entities/observers/societies have had the capacity to satisfy the three conditions discussed above. This assumption would be invalidated if the “Silurian hypothesis” of ref. 23 were confirmed, which considers the possible existence of industrial civilizations predating humanity (24), in which case we would certainly advocate revisiting the calculations that are described in this paper.

The emergence of human civilization could be defined in a variety of ways. Some possible defining “moments” could be the appearance of hominids, the evolution of Homo sapiens, complex language, the Neolithic revolution, or the first radio transmissions into space. Whatever we use, this shifts tI around only by several million years at most. Since tI is of order of several gigayears, these disagreements have negligible impact on our final results and thus tI will be treated as a fixed quantity.

This is not true for the first transition, since it seems to have occurred relatively quickly and the uncertainty associated with it is comparable to the actual timing. Further, it is not possible to accurately date the emergence of life, since any life could (and indeed must) have begun prior to its appearance in the geological record. Accordingly, the true date for the emergence of life, tL, must predate the actual observation, tL′; i.e., tL<tL′. The probability of this can be calculated through integration:Pr(tL<tL′,tI|λL,λI)=∫tL=0tL′Pr(tL,tI|λL,λI)dtL.[5]Since tI is defined as the time since tL, then one cannot directly measure this value either. However, we can state that its value is somewhere between tI′ and tI′+tL′, where tI′ is the observed time difference between the emergence of intelligence and the first evidence for life. This allows us to write our final likelihood function asL=Pr(tL<tL′,tI′<tI<tI′+tL′|λL,λI)  =∫tI=tI′tI′+tL′Pr(tL<tL′,tI|λL,λI)dtI.[6]Evaluating the above yields a piecewise closed-form likelihood function which is given in SI Appendix. The function has two subdomains, one which applies to the interval T>2tL′+tI′ and one which applies to T<2tL′+tI′. As shown later, the former case is applicable when tL′<0.904 Gy and we plot this function in Fig. 1 against λL and λI, along with the limiting behaviors (SI Appendix).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 1.

The joint-likelihood function for the rate of abiogenesis, λL, and the rate of intelligence emergence, λI (for cases where T>2tL′+tI′). Contours relative to the maximum-likelihood position (gray circle) are depicted by gray lines, where one can see a preference against low λL values. The limiting behaviors of the likelihood function are shown along Top, Bottom, Left, and Right edges.

It is interesting to note that the likelihood function is not monotonic and has a global maximum which can be solved for numerically. For example, using the “optimistic” data defined in Adopted Values for the Observational Data, it occurs at λ^I→0 and λ^L=21.8 Gy−1 (the full behavior is shown in SI Appendix). However, along the λI→0 axis, the likelihood is almost flat beyond this peak. For example, with the same data, the likelihood is 52.7% of the peak when one sets λL=λ^L/10, but only 99.4% of the peak when one sets λL=10λ^L. While maximum-likelihood parameters are instructive, we turn to Bayesian inference to determine the posterior distributions and rigorously compare different model scenarios.

Adopted Values for the Observational Data.

To perform any inference with our likelihood function, one first needs to assign values to the observables tL′ and tI′, as well as T. All three times are relative to some initial time when conditions on Earth became suitable for life to emerge, and so let us first discuss how to define this initial time.

There is of course uncertainty about the conditions on the early Earth and when they became suitable for life (25⇓–27). Earth is generally thought to have been impacted by a Mars-sized body, dubbed “Theia,” 4.51 Gy ago in a cataclysmic event that formed the Moon (28). Such an impact would have been a globally sterilizing event and indeed may have been accompanied by another sterilizing impactor, “Moneta,” 40 My later (29). Mineralogical evidence from zircons indicates that both an atmosphere and liquid water must have been present on Earth’s surface (4.404±0.008) Gy ago (30). In this work, we consider these to be the necessary basic requirements for abiogenesis to take place and thus adopt T=4.408 Gy throughout. As noted by ref. 13, a step-function–like transition from uninhabitable to habitable is surely too simplistic, but our ignorance of Earth’s early history and indeed the conditions necessary for life mean that we do not at present have a well-motivated complex model to impose in its place.

The earliest evidence for life arrives soon after this time, in the form of ¹³C-depleted carbon inclusions within 4.1-Gy-old zircon deposits (17). The source of this depletion remains controversial but this would yield an optimistic estimate of tL′=0.304 Gy. The earliest direct and undisputed evidence for life comes from microfossils discovered in 3.465-Gy-old rocks in western Australia (15, 16, 18), yielding tL′=0.939 Gy. We highlight that, in both the optimistic and the conservative case, tL′ is much larger than the 76-My uncertainty as to when Earth became habitable and thus is not a dominant source of uncertainty.

For tI′, for reasons discussed in Accounting for Observational Constraints we can simply attribute this time to be the modern era. We therefore adopt intelligent observing as arriving “now” such that tI′=4.404 Gy−tL′ in what follows.

Finally, we turn to T, which defines the interval over which Earth is expected to persist as habitable. It is important that we refine this definition to be habitable for intelligent beings, such as ourselves. If Earth evolves into a state where only simple microbial life is possible, then a success can no longer occur for process I. As the Sun evolves, its luminosity will increase, which in turn increases the rate of weathering of silicate rocks on Earth (31). This increased weathering draws down carbon from the atmosphere, thus gradually depleting the atmospheric content of carbon dioxide. Once levels drop below ∼10 ppm, plant C4 photosynthesis will no longer be viable (32), leading to their imminent demise. It is also possible that higher temperatures and the progressive loss of Earth’s oceans could trigger an earlier die-off (33).

The end of plant life leads to a collapse in both the food chain and Earth’s oxygen productivity, upon which animal life is critically dependent. Large endotherms, such as mammals and birds, will be the first to become extinct as a result of their higher oxygen requirements (31, 34). Thus, one can reasonably consider that the habitable window for intelligence decidedly ends once the reign of plant life comes to a close. The timing for this is predicted to be 0.9 Gy by ref. 33, a value which is adopted in what follows to give T=5.304 Gy.

In summary, we set T=5.304 Gy but consider two values for tL′ of tL′=0.304 Gy (“optimistic”) and tL′=0.939 Gy (“conservative”). This in turn gives two values of tI′ of tI′=4.404 Gy−tL′.

The Role of the Prior.

Equipped with a likelihood function, one may infer the a posteriori distribution of λL and λI using Bayes’ theorem:Pr(λL,λI|tL′,tI′)=Pr(tL′,tI′|λL,λI)Pr(λL,λI)∬Pr(tL′,tI′|λL,λI)Pr(λL,λI) dλL dλI.[7]In any Bayesian inference problem, the posterior is a product of the likelihood and the prior and thus is affected by both. In cases where one possesses little or no information about the target parameters in advance, such as here, the ideal prior should be minimally informative (“diffuse”) such that it does not strongly influence the result (35). In objective Bayesianism, the resulting posterior should be expected to be universally agreed upon by everyone (36)—whereas a subjective Bayesian would argue that probability corresponds to the degree of personal belief (37).

When equipped with strongly constraining data, even those using different diffuse priors will generally find convergent solutions, since the likelihood overwhelms the prior, thus naturally leading to objective Bayesianism. This is certainly not the case for our problem, since it has already been established that the posterior for λL is very sensitive to the priors (13). In such a case, one should tread carefully and seek a prior which can be objectively defined, such that other parties could agree upon the choice of prior and thus the resulting posterior.

We therefore proceed by considering how to define a prior distribution which is minimally informative and also not dependent upon subjective choices of the prior distribution parameters. However, we will later show that several important inference statements can be made independent of the prior.

Power Law in λ.

In previous work (13, 22), the a priori distribution for λL was assumed to be of the form λLn—a power law. In extending the likelihood to include λI, one may similarly extend this power-law prior to encompass λI by writing that Pr(λL,λI)=Pr(λL)Pr(λI) (i.e., assuming independence), wherePr(λ)=λ−1log(λmax)−log(λmin) if n=−1,(n+1)λnλmaxn+1−λminn+1 otherwise.[8]For n=0, this returns a uniform in λ prior, for n=−1 a uniform in log⁡λ prior, and for n=−2 a uniform in λ−1 prior: the three priors considered by ref. 13. In adopting a prior of this form, one must choose values for three shape parameters: the index, n and the prior bounds λmin and λmax.

Assigning the Prior Shape Parameters.

In ref. 13, the favored index was n=−1 on the basis that a log-uniform prior exhibits scale-invariance ignorance for λ. For a real-valued parameter constrained only by a minimum and a maximum threshold, n=−1 also corresponds to the Jeffreys prior—a standard approach to defining objective priors (35). However, in this case, the n=−1 power law is not actually objective since one cannot objectively define a minimum and a maximum threshold. Because a power-law prior does not have semiinfinite support from λ=0 to λ=∞, then these prior bounds have to be subjectively chosen.

For these bounds, ref. 13 set λmax=103 Gy−1 somewhat arbitrarily and a range of plausible values were offered for λmin. While useful as an exercise to test the sensitivity of the posterior to the prior, this approach does not enable an objectively defined solution. In an effort to objectively assign a power-law prior, we consider here imposing the condition that the prior should be fair and unbiased, which we define in what follows.

As currently stated, the prior in Eq. 8 appears reasonably diffuse for n=0, −1, and −2, and the bounds could essentially be anything. However, it is worth recalling that the Poisson model is used as a vehicle to describe the Bernoulli probability of one or more successful events occurring. Accordingly, a natural alternative parameterization for this problem is to consider the fraction of experiments in which the Poisson processes culminate in at least one success, fL and fI. Although these terms are similar to the fractions defined in the Drake equation (38), here an “experiment” really refers to rerunning Earth’s history back and observing how the stochastic processes play out each time (rather than some other world). Since f=1−e−λT, then the power-law prior in λ is transformed into f space asPr(f)=n+11−f(−log(1−f))n(−log(1−fmax))n+1−(−log(1−fmin))n+1.[9]Recall that we seek to define a prior which is both fair and unbiased. For a Bernoulli process, such as a coin toss, a “fair” prior can be defined as one for which the chance of a positively loaded coin is no more or less likely than that of a negatively loaded one. Accordingly, in our problem, we define a fair prior as one which does not a priori favor either an optimistic (f>1/2) or a pessimistic (f<1/2) worldview. This can be quantified by defining the prior odds ratio between the two scenarios using F:F≡∫f=1/2fmaxPr(f)df∫f=fmin1/2Pr(f)df.[10]Setting F=1, one may solve for fmax (which corresponds to λmax) as a function of fmin (corresponding to λmin):limF→1λmax=(log⁡2)2T2λmin if n=−1,(2⁡log⁡2)n+1−(λminT)n+11/(n+1)T otherwise.[11]Although this ensures a fair prior (subject to our definition), it does not necessarily ensure an unbiased one. An unbiased prior is defined here as one for which the a priori expectation value of f, given by E[f]≡∫fminfmaxfPr(f)df, equals one-half. After imposing the F=1 constraint enabled by Eq. 11, we evaluate E[f] in the limit of fmin→0 as a function of n. This reveals for n=−1 that the expectation value converges to one-half, as desired for an unbiased prior (SI Appendix). The only other value of n in the range −1<n<2 that yields a fair and unbiased distribution is n=−0.709…, but this is shown in SI Appendix to require an overly restrictive prior bound limit and thus is not used.

Using these two constraints, our fair and unbiased prior for λ takes the form λ−1, with bounds following the relationship given by Eq. 11. Unfortunately, our two constraints applied to three parameters are insufficient to uniquely define the prior—it is still necessary to choose λmin subjectively. The posterior could still be argued to be objective if it were found to be broadly insensitive to the choice of λmin—however, this unfortunately turns out to be false, as shown in what follows.

Fair and Unbiased Priors with Semiinfinite Support.

Since we have no way of objectively choosing λmin, then a fair and unbiased (subject to our definitions) power-law prior in λ does not provide a viable path to defining an objective posterior.

Rather than prescribing a prior in λ space and then evaluating its fairness and bias in f space, we consider here simply writing down a fair and unbiased prior in f space directly. Since f represents a fraction, it can be interpreted as the Bernoulli probability of a success over the interval T. For a Bernoulli process, a so-called Haldane prior (39) of the form ∝(f(1−f))−1 was argued by the objective Bayesian Jaynes (36) to represent the least informative prior. Such a prior is fair and unbiased by construction and places extreme weight on the solutions f=1 and f=0 at the expense of intermediate values. The intuition behind this is that either a very small fraction of planets will be successful or almost all of them will be, but it is unlikely the laws of nature are tuned such that approximately half of the planets are successful.

Using the Fisher information matrix, one can define the Jeffreys prior for a Bernouilli distribution. The solution is not the Haldane prior, but rather a softer variant of the form ∝(f(1−f))−1/2. This translates to a prior in λ space given byPr(λ)=TπeλT−1.[12]Both the Haldane and Jeffreys priors are fair and unbiased with respect to f and indeed any prior of the more general form ∝(f(1−f))n satisfies these conditions. However, the n=−1 case, corresponding to the Haldane prior, is improper and indeed leads to an improper posterior too, but for all other n>−1 the prior can be normalized to a finite quantity. Accordingly, the Jeffreys prior is fair, unbiased, objectively defined, and proper over the interval f=[0,1]. Since f=0 corresponds to λ=0 and f=1 corresponds to λ=∞, this naturally yields a proper prior with semiinfinite support in λ, something which was not possible with the power-law case discussed earlier. Together, these properties make the distribution well suited for our problem and we argue it solves the dilemma faced in earlier work (13).

Although we consider the Jeffreys prior to be the ideal objective prior for our problem, it is instructive to consider posteriors with n=0 (a uniform prior in f) as well, which has a λ-space form ofPr(λ)=Te−λT.[13]

Bayes Factors Independent of the λ Prior.

Equipped with our likelihood function and prior, one may now sample/integrate the posterior probability distribution to compute marginalized distributions. Marginalization irreversibly bakes the prior into the resulting collapsed posteriors, but specific probabilistic statements can be made in a Bayesian framework without marginalizing. In particular, we consider here an exercise in Bayesian model comparison, where we seek to compare four models, ℳ, defined as the unique corners of the parameters volume:

• • ℳ00: λL≪1/tL′ and λI≪1/tI′

• • ℳ01: λL≪1/tL′ and λI≫1/tI′

• • ℳ10: λL≫1/tL′ and λI≪1/tI′

• • ℳ11: λL≫1/tL′ and λI≫1/tI′.

Binarizing the parameter volume into these four camps may at first seem arbitrary—What about intermediate values? However, this partitioning is consistent with objective Bayesianism. The objective Bernoulli prior treats life/intelligence as being either very rare or very common, but unlikely to be finely tuned such that it approaches the intermediate value of one-half—thus motivating the models above.

Conditioned upon some available data, D, one may express the odds ratio between two models as Pr(M1|D)/Pr(M2|D)=[Pr(D|M1)/Pr(D|M2)][Pr(M1)/Pr(M2)]. The terms inside the first square bracket are known as the Bayes factor, which equals the odds factor under the simple assumption that no model is a priori preferred over any other. The Bayes factor is the ratio of two “evidences” given by Z≡Pr(D|M), and for the four models defined above, Z can be expressed analytically and independent of the prior π(λL,λI). This can be seen by noting that, for example with model M00:Z00=Pr(tL′,tI′︷D|λL≪1/tL′,λI≪1/tI′︷M00)  =limλL≪1/tL′limλI≪1/tI′Pr(tL′,tI′|λL,λI)︸=L.[14]Thus, the evidences of these four corner models are independent of the prior of π(λL,λI), meaning that even those adopting different priors would consistently agree on the Bayes factors. We note that ref. 13 used a similar strategy and we provide an alternative explanation of this prior-free model comparison in SI Appendix, in terms of the Savage–Dickey ratio (40).

In practice, Bayes factors for the two corners with rapid intelligence emergence (λI≫1/tI′) tend to zero, since this is the behavior of the likelihood function (SI Appendix). This can be understood by the fact that if the rate of intelligence emergence were extremely fast, then it would be incompatible with taking as long as it did here on Earth. In contrast, life could emerge much faster than tL′ because tL′ represents only the first appearance of life in the geological record, not the actual date of abiogenesis.

Since these two corners are zero, we instead compare models M10 to M00. This represents the Bayes factor between a scenario where abiogenesis is a fast versus a slow process conditional upon the premise that intelligence emergence is itself a slow process. In this case, we findZ10Z00=T2tL′ if T>2tL′+tI,TtL′4(T−tI′)tL′−2tL′−(T−tI′)2 if T<2tL′+tI,[15]which evaluates to (Z10/Z00)=8.73 and 2.83 for the optimistic and conservative data inputs, respectively. The above also reveals the dependency on T is nearly linear; if T is revised significantly up, then optimism for life would also increase. We highlight that Z10<Z00 if tL′>3.72 Gy—i.e., if the earliest evidence for life were from no earlier than 680 My ago, we would conclude that abiogenesis was an improbable event.

If we relax the assumption that λI≪1/tI′ and let intelligence become faster, then the Bayes factor monotonically rises, as shown in Fig. 3. This therefore means that the Bayes factor of a quick versus slow abiogenesis scenario must be greater than the limiting case of Z10/Z00, irrespective of whatever value λI takes (or indeed whatever the prior is).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 3.

(A) Bayes factor for a model where life emerges rapidly (λL≫1/tL′) versus slowly (λL≪1/tL′) on Earth. (A) A quick start is favored by at least a factor of 3 conditioned upon early microfossil evidence, independent of our assumptions regarding the evolutionary timescale of intelligent observers and priors on the abiogenesis rate. (B) Bayes factor of a scenario where intelligent observers typically emerge on a much longer timescale than occurred on Earth, versus the ensemble of possibilities. There is a weak preference for a rare intelligence scenario.

On this basis, we can conclude that even with the most conservative date for the emergence of life, a scenario where abiogenesis occurs rapidly is at least three times more likely than a slow emergence, independent of the priors and even the timescale it takes for intelligence to emerge. If the more ambiguous evidence for an earlier start to life is confirmed (17), then this would increase the odds to a factor of 9, representing relatively strong preference for a model where life would consistently emerge rapidly on Earth, if time were replayed.

Bayes Factors after Marginalization.

Thus far we have avoided using the marginalized posteriors, which has the benefit of enabling model comparison independent of the prior π(λL,λI). However, it also has the disadvantage that we can compare only conditional scenarios. For example, our result for (Z10/Z00) is a Bayes factor conditional upon the assumption of a slow intelligence emergence. While it turns out this can be interpreted as a lower limit on a fast versus slow abiogenesis scenario, marginalization allows for a calculation which integrates over the uncertainty in λI. For example, one may write that the evidence for a model where λL≪1/tL′ (slow abiogenesis) marginalized over λI is given byZL−slow=∫λI=0∞limλL≪1/tL′Lπ(λI)dλI.[16]We numerically evaluated the evidences for ZL−slow using the above, ZL−slow (λL≫1/tL′) as well as ZI−slow (λI≪1/tI′) and the ensemble evidence over all possibilities, Zensemble. As before, the fast intelligence emergence scenario has zero evidence since the likelihood tends to zero in this regime, for reasons discussed earlier. The resulting Bayes factors for (ZL−fast/ZL−slow) are shown in Fig. 3A by the solid (Jeffreys prior) and open (uniform prior) circles.

The two sets of points are almost indistinguishable and consistently lie above our previously derived lower limit on the Bayes factor, as expected. Using the optimistic data, we find that (ZL−fast/ZL−slow)=9.538 and (ZL−fast/ZL−slow)=9.648 for the Jeffreys and uniform priors, respectively, both of which satisfy (ZL−fast/ZL−slow)>(Z10/Z00)=8.73. For the conservative data, these numbers become (ZL−fast/ZL−slow)=3.110 and (ZL−fast/ZL−slow)=3.137, again satisfying (ZL−fast/ZL−slow)>(Z10/Z00)=2.83.

Further understanding of these Bayes factors can be gained by evaluating the marginalized posteriors. We numerically marginalize the posteriors in the case of the optimistic data and show the resulting distributions in Fig. 4. From these, one can see that the fL→0 limit drops below the prior, whereas the fL→1 limit rises above it. Together, these results paint a consistent picture that the timing of life’s emergence and that of intelligent observers favor the hypothesis that life would likely reemerge rapidly on Earth were the clock to be rerun.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 4.

(Top row) Marginalized posteriors (solid lines) for λL (Left), λI (Center Left), fL (Center Right), and fI (Right) using the objective Bernoulli prior (dashed lines), using the optimistic data. (Bottom row) Same as Top row but using the uniform prior in f for comparison. The y-axis scale is chosen to highlight the dynamic range.

And What of Intelligence?

Thus far, we have calculated Bayes factors concerning fast versus slow abiogenesis rates. For the rate of emergence of intelligent observers, Bayes factors against a fast emergence scenario tend to infinity, since limλI≫1/tI′L→0 (as shown in SI Appendix). Instead, it is more useful to compare the slow intelligence scenario (λI≪1/tI′) against the ensemble of models where λI can take any value, for which the Bayes factor isZI−slowZensemble=∫λL=0∞limλI≪1/tI′L(λL,λI)π(λL)dλL∫λL=0∞∫λI=0∞L(λL,λI)π(λL)π(λI)dλIdλL.[17]We evaluated the above numerically using optimistic data to yield 1.638 and 1.474 using the Jeffreys and uniform priors, respectively. Switching to the conservative data points barely affects these numbers with 1.572 and 1.416 (the full range of possibilities is depicted in Fig. 3B). This suggests a slight preference, 3:2 betting odds, that intelligent observers would rarely reemerge—a value which is broadly robust against the two priors considered and the range of possible tL′ values.