On the Rate of Abiogenesis from a Bayesian Informatics Perspective (1806.08033v1)

Abstract: Life appears to have emerged relatively quickly on the Earth, a fact sometimes used to justify a high rate of spontaneous abiogenesis ($\lambda$) among Earth-like worlds. Conditioned upon a single datum - the time of earliest evidence for life ($t_{\mathrm{obs}}$) - previous Bayesian formalisms for the posterior distribution of $\lambda$ have demonstrated how inferences are highly sensitive to the priors. Rather than attempt to infer the true $\lambda$ posterior, we here compute the relative change to $\lambda$ when new experimental/observational evidence is introduced. By simulating posterior distributions and resulting entropic information gains, we compare three experimental pressures on $\lambda$: 1) evidence for an earlier start to life; $t_{\mathrm{obs}}$; 2) constraints on spontaneous abiogenesis from the lab; and 3) an exoplanet survey for biosignatures. First, we find that experiments 1 and 2 can only yield lower limits on $\lambda$, unlike 3. Second, evidence for an earlier start to life can yield negligible information on $\lambda$ if $t_{\mathrm{obs}} \ll \lambda_{\mathrm{max}}^{-1}$. Vice versa, experiment 2 is uninformative when $\lambda_{\mathrm{max}} \gg t_{\mathrm{obs}}^{-1}$. Whilst experiment 3 appears the most direct means of measuring $\lambda$, we highlight that early starts inform us of the conditions of abiogenesis, and that lab experiments could succeed in building new life. Together then, the three experiments are complimentary and we encourage activity in all to solve this grand challenge.

• The paper demonstrates how Bayesian informatics and Kullback-Leibler Divergence quantify the learnable information on abiogenesis rate (λ) from experimental data.
• It shows that revising early evidence and tightening lab constraints yield probabilistic lower limits on λ, with saturation effects when prior bounds dominate.
• The exoplanet biosignature survey is highlighted as a promising method, capable of directly measuring λ when intermediate detection rates are observed.

This paper (1806.08033) addresses the challenge of inferring the rate of abiogenesis ($\lambda$) on Earth-like planets, which is severely constrained by having only one known example of life: life on Earth, which appears to have started relatively early. The authors build upon previous Bayesian frameworks that showed the resulting probability distribution of %%%%1%%%% (the posterior) is highly sensitive to the initial assumptions (the prior). Instead of trying to infer the absolute value of $\lambda$, the paper focuses on quantifying how much we can learn about $\lambda$ from potential future observations and experiments, using a Bayesian informatics approach and the Kullback-Leibler Divergence (KLD) as a measure of information gain.

The core model assumes abiogenesis events follow a Poisson process with rate $\lambda$. The time for life to emerge ($t_{\text{life}}$) on a planet then follows an exponential distribution $\mathcal{E}(\lambda)$. To account for the fact that we, as intelligent observers, exist, the model includes a selection bias: life must have emerged early enough on Earth to allow for the evolution of intelligent observers within the planet's habitable lifetime. This is modeled by truncating the exponential distribution at a time $\tau$, representing the latest possible time for life's emergence compatible with our existence. The observed earliest evidence for life on Earth ($t_{\text{obs}}$) provides a lower bound on $t_{\text{life}}$, meaning $t_{\text{life}} \leq t_{\text{obs}}$. Combining these constraints, the likelihood function for $\lambda$ given $t_{\text{obs}}$ and $\tau$ is $$\mathrm{Pr}(t_{\text{obs}} | \lambda, \tau) = \frac{1-e^{-\lambda t_{\text{obs}}}}{1-e^{-\lambda \tau}}$$.

For the priors, the paper uses a log-uniform distribution for $\lambda$ between $\lambda_{\text{min}}$ and $\lambda_{\text{max}}$ and a uniform distribution for $\tau$ between 1.5 Gyr (minimum time for intelligent life evolution) and 4.5 Gyr (Earth's approximate habitable age). The posterior distribution of $\lambda$ is then sampled using a bootstrap particle filter.

The paper analyzes the potential information gain from three hypothetical experiments:

1. Experiment 1: Revising $t_{\text{obs}}$ (Paleontology/Geology): This involves finding earlier evidence for life on Earth, effectively reducing the value of $t_{\text{obs}}$.
   • Practical Implication: Continued paleontological and geological studies aiming to push back the date of the earliest life.
   • Findings: Revising $t_{\text{obs}}$ to earlier times generally favors higher values of $\lambda$, but the resulting posterior remains a monotonic function, providing only a probabilistic lower limit on $\lambda$, not a precise measurement. The information gain saturates when $t_{\text{obs}}$ becomes significantly smaller than $\lambda_{\text{max}}^{-1}$. This means that if the actual rate $\lambda$ is very high (and thus $\lambda_{\text{max}}$ is also high), pinpointing the exact earliest moment of life on Earth beyond a certain point provides diminishing returns for constraining $\lambda$ itself (though it remains crucial for understanding the conditions for abiogenesis).
2. Experiment 2: Reducing $\lambda_{\text{max}}$ (Laboratory Experiments): This involves conducting extensive laboratory experiments simulating early Earth conditions and observing whether life emerges spontaneously. Null results from these experiments can be used to set an upper limit on the rate of spontaneous abiogenesis under those specific conditions, which can inform $\lambda_{\text{max}}$.
   • Practical Implication: Conducting numerous Miller-Urey-type experiments or more advanced simulations of abiogenesis.
   • Findings: A tighter upper limit on $\lambda$ derived from null results effectively truncates the $\lambda$ posterior at lower values. Like Experiment 1, this typically results in a posterior that provides a lower limit on $\lambda$. The information gain in this experiment can be significant, especially if $\lambda_{\text{max}}$ is constrained to values comparable to or lower than $t_{\text{obs}}^{-1}$. The information gain saturates when $\lambda_{\text{max}}$ is significantly larger than $t_{\text{obs}}^{-1}$. The paper notes that a successful, reproducible lab abiogenesis event (not just a null result) would be dramatically more informative but was excluded from the analysis as it would make the comparison moot.
3. Experiment 3: Exoplanet Survey for Biosignatures: This involves surveying $N$ Earth-like exoplanets and determining if life is present on $M$ of them. The probability of a positive detection on a single planet is modeled as $p = 1 - e^{-\lambda t_{\text{planet}}}$, where $t_{\text{planet}}$ is the age of the planet (assumed to be 5 Gyr for simplicity). The likelihood for detecting $M$ planets out of $N$ follows a Binomial distribution.
   • Practical Implication: Developing and using powerful telescopes (like future space telescopes) capable of detecting biosignatures in exoplanet atmospheres.
   • Findings: This experiment has the potential to yield a peaked $\lambda$ posterior, providing a direct measurement of $\lambda$ rather than just a limit, unlike the other two experiments. The shape and peak of the posterior depend strongly on the observed success rate $M/N$ and the sample size $N$.
     • Counter-intuitive result: If the observed success rate $M/N$ is very high (e.g., $M \approx N$), it might be less informative than detecting life on only a few planets or none at all, depending on the prior. This is because, with certain choices of the $\lambda$ prior (especially those allowing for high rates, as suggested by Earth's early life), our prior expectation might already be that most Earth-like planets have life. Confirming this expectation yields less information gain than a surprising result (like finding life is rare).
     • Detecting $M=0$ (no life) leads to a posterior peaked at $\lambda_{\text{min}}$. Detecting $M=N$ (life on all surveyed planets) leads to a posterior peaked at $\lambda_{\text{max}}$. Intermediate success rates ($0 < M < N$) tend to produce peaked posteriors away from the prior boundaries, representing a stronger measurement.
     • The informativeness of this experiment is highly sensitive to the choice of $\lambda_{\text{min}}$, which cannot be constrained empirically by the other experiments.

Summary Comparison:

The paper concludes that the three experiments are complementary:

• Experiments 1 (earlier $t_{\text{obs}}$) and 2 (tighter $\lambda_{\text{max}}$ from null lab results) primarily constrain $\lambda$ by setting probabilistic lower limits. They are limited by saturation effects dependent on the prior range ($\lambda_{\text{max}}$ and $t_{\text{obs}}$ respectively).
• Experiment 3 (exoplanet survey) has the potential to provide a direct measurement of $\lambda$ (a peaked posterior) if an intermediate success rate ($0 < M < N$) is observed. However, its informativeness is highly dependent on the prior expectation (influenced by Earth's history and the chosen $\lambda_{\text{min}}$).
• Even if the information gain on $\lambda$ is small in certain scenarios (e.g., finding life on all surveyed exoplanets), these experiments still provide crucial information about the conditions under which life arises (paleo/lab) and the diversity of inhabited environments (exoplanets).

The paper highlights the limitations of the analysis, including simplifying assumptions about a universal $\lambda$, constant environment over time, and unambiguous detection in exoplanet surveys. Future work could incorporate hierarchical models and softer detection probabilities. Ultimately, pursuing all three research directions is valuable for addressing the grand challenge of understanding abiogenesis.