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A Bayesian Analysis of the Astrobiological Implica6ons of the Rapid Emergence of Life on the Early Earth*
Princeton University, Department of Astrophysical Sciences and University of Tokyo, IPMU
++++++++++++++++++++++++++++

Ed Turner

Harvard University, Sta0s0cs Department ­ November 15, 2011 *CollaboraCon with David Spiegel (Princeton University Observatory) (+ thanks to former Princeton undergraduate Carl BoeJger '07)


Preface
· Field: Astrobiology · Content: Mostly staCsCcs with small amounts of biology & geology but almost no astronomy · MoCvaCon 1: Clarify the implicaCons of the early appearance of life on Earth for biology elsewhere · MoCvaCon 2: A worked example of a Bayesian approach to a hand waving, intuiCve, ambiguous, judgment call, small number sta0s0cs problem with some parallels to anthropic arguments


Fundamental Ques7ons of Astrobiology
· Does extraterrestrial life exist? · What is its nature? · How common is it? The overwhelmingly most pracCcal and promising approaches to these quesCons are empirical, searches for life on bodies in the Solar System and beyond. But that is not our topic today.


A Possible in Principle Calcula7on
Given Pabiogenesis(physical condi0ons) per unit volume and per unit Cme, and prevalence of such suitable abioCc enviroments in the Universe a straigh\orward staCsCcal esCmate could be made. But, alas, we are quite ignorant of both and have no immediate prospects of remedying either situaCon.


A Poten7al Finesse
· Exoplanet studies strongly suggest, but have not yet proven, that planets resembling the Earth in a very general/crude way (mass, primary star, orbit, composiCon etc) are reasonably common. · Life arose very quickly on the early Earth. · This suggests that Pabiogenesis(early Earth like condi6ons) per unit planet per unit age of the Earth is not extremely small. · Thus, simple/primiCve extraterrestrial life (at least) is not too rare. roughly the current "consensus" view


How quickly is very quickly?
· Earth formed at 4.54 Ga (Ga = 109 yrs ago, like redshie) · Water was reasonably abundant by ~4.3 Ga · Likely sterilizaCon by the LHB and subsequent global volcanic resurfacing around 3.85 4.0 Ga · Highly controversial isotopic indicators (13C depleCon) of metabolic acCvity in the oldest known surviving rocks, 3.7 3.82+ Ga · Wide spread "probable fossils" of microbes & macroscopic microbial biofilms (stromatolites) in sediments formed ~3.5 Ga · Definite, highly evolved macro fossils ~3.2 Ga


Stromatolites


(Van Zuilen 2006)


Abiogenesis almost certainly occurred on Earth significantly earlier than the Cme of the most ancient life detected to date. Thus, "very quickly" must mean within a few hundred million years but it could have been much faster.


How small is extremely small? (1)
· The basic molecular chemistry of all terrestrial life is essenCally idenCcal and very complex. · Two classes of macro molecules, proteins and nucleic acids, play central roles currently, but it is imagined that RNA alone might have sufficed for an earlier form of life (the "RNA world"). · The building blocks of both proteins (amino acids) and nucleic acids (nucleoCdes) are a set of small molecules which form bonds in an arbitrary order to create the long polymers which are these macro molecules.


How small is extremely small? (2)
· Both amino acids and nucleoCdes are produced in reasonable abundance by physical (abioCc) chemical reacCons that plausibly (empirically in some cases) commonly occur in nature (GOOD!). · However, the polymers in quesCon are huge, typically 100s to 100s of millions of "building blocks" long, with even minimally funcConal ones requiring ~100 nucleoCdes or amino acids · The probability of any specific one arising randomly is then factorially small, i.e, 10 100s or less (BAD!). · Thus, "extremely small" no extraterrestrial life


Abiogenesis: StochasCc Lego ConstrucCon?


Ward & Brownlee (2000) RARE EARTH
"the Cme from soup to bugs may have been far less than 10 million years. Making life may be a rapid operaCon ­ a key observaCon supporCng our contenCon that life may be very common in the Universe."


Irra7onal Exuberance
"Personally, given the ubiquity and propensity of life to flourish wherever it can, I would say that, my own personal feeling is that the chances of life on this planet are 100%. I have almost no doubt about it." Steven Vogt speaking to L. J. Zgorski (NSF) in regard to Gl 581g, a reported exoplanet with Msin(i)3M orbiCng within its primary's HZ


Sensible (but not exactly right)
"It is too glib to claim that if the origin of life took place on Earth immediately aeer the end of the heavy bombardment, then the origin of life must be `easy,' so that the prospects for life elsewhere are increased" Chyba & Hand (2005) ARAA, 43 Astrobiology: The Study of the Living Universe


N.B. ­ A formal frequen0st analysis with a misleading result


Start from the Bayesian's F=ma
A = the model B = the data P[A|B] = probability of the model given the data = the posterior (what we want!) P[B|A] = probability of the data given the model = the likelihood (what we can obtain from a "forward" calculaCon) Pprior[A] = a priori probability the model is true/correct = the prior (oeen requires "arbitrary" judgment calls, source of controversy) P[B] = probability of the data (typically unknown, but only needed for normalizaCon)


A Uniform Rate (Poisson) Model
= rate of abiogenesis per Gyr per Earth like planet t = age of the Earth (like expansion factor) sterile = earliest Cme at which life could develop max = latest Cme at which life could develop n = number of independent abiogenesis events


Two "Anthropic" Constraints
· Life must have arisen on Earth one Cmes or we would not be here to carry out the calculaCon

· Life must have arisen early enough on Earth to allow us to evolve by the present temerge = Earth's age at abiogenesis, t0 = its present age & evolve = Cme for evoluCon of astrobiologists


The Likelihood Term

where In the limit of extremely small

In the limit of extremely large

P[B|A] 1


The Bayes Factor/Ra7o/Evidence for Model Selec7on

where

Rule of thumb: R < 10 barely worth menConing


The Prior Factor ­ Problema7c!
Uniform: Informa0ve=BAD Inverse Uniform: Informa0ve=BAD Log Uniform: Uniforma0ve=GOOD but improper=not so good
max = 1000 Gyr 1 L&D2002 equivalent


Use funcCon esCmates for the f terms



HypotheCcal = extremely quick terrestrial abiogenesis (speculaCve) ConservaCve = slow terrestrial abiogenesis consistent with the data OpCmisCc = quick terrestrial abiogenesis suggested by the data


Turn the Bayes crank posterior distribuCons


10 10 10 Probability Density 10 10 10 10 10 10

1

0

Optimistic
1

2

3

4

5

Uniform Log ( 3) InvUnif ( 3) Posterior: Solid Prior: Dashed

6

7

3

2

1

0 1 1 log10[ ] ( in Gyr )

2

3


1 0.9 0.8 Cumulative Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3 2 1 0 1 1 log10[ ] ( in Gyr ) 2 3 Uniform Log ( 3) InvUnif ( 3) Posterior: Solid Prior: Dashed

Optimistic


1 0.9 0.8 Cumulative Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 22 19 16 13 11 8 1 log10[ ] ( in Gyr ) 5 3 0 3 Posterior: Solid Prior: Dashed

Optimistic
Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22)


2 0 Bounds on log10[ ] ( in Gyr ) 2 4 6 8 10 12 14 16 18 20 22 20 Median Value of 1 2 Lower Bound Lower Bound 0 25 50 75
1

Optimistic
( 18
max

= 103 Gyr 1) 16 14 log10[

100 100

80 8

60

40 6

20 4

0 2

12 10 1 ] ( in Gyr ) min


1 0.9 0.8 Cumulative Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 22 19 16 13 11 8 1 log10[ ] ( in Gyr ) 5 3 0 3 Posterior: Solid Prior: Dashed

Hypothetical
Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22)


1 0.9 0.8 Cumulative Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 22 19 16 13 11 8 1 log10[ ] ( in Gyr ) 5 3 0 3 Posterior: Solid Prior: Dashed

Conservative
Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22)


Summary
· Paleobiological & geological evidence favors a very fast development of life on the early Earth. · If so, this datum implies a best esCmate value of very roughly of order ~1 Gyr 1 or greater. · However, even extremely small values of , that nearly exclude life elsewhere in the observable universe, are consistent with the datum. · Stronger intuiCve or formal conclusions about are the consequence of an informaCve prior and not an implicaCon of the datum.


Conclusions
A Bayesian fan of extraterrestrial life should be encouraged by, but not highly confident based on, the rapid emergence of life on the early Earth. A straigh\orward, but careful, Bayesian analysis can yield transparent and well founded results even in situaCons involving (very) small number staCsCcs, anthropic consideraCons, intuiCve arguments, uncertain data etc.





1 0.9 0.8 Cumulative Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3 2 1 0 1 1 log10[ ] ( in Gyr ) 2 3 Mars, logarithmic prior Posterior: Solid Prior: Dashed


10 10 10 Probability Density 10 10 10 10 10 10 10

0

1

2

Mars, logarithmic prior

3

Posterior: Solid
4

Prior: Dashed

5

6

7

8

9

3

2

1

0 1 1 log10[ ] ( in Gyr )

2

3


10

0

10 Probability Density

5

10

10

Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22) Posterior: Solid Prior: Dashed

10

15

10

20

Optimistic
10
25

22

19

16

13 11 8 1 log10[ ] ( in Gyr )

5

3

0

3


10

0

10 Probability Density

5

10

10

Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22) Posterior: Solid Prior: Dashed

10

15

10

20

Hypothetical
10
25

22

19

16

13 11 8 1 log10[ ] ( in Gyr )

5

3

0

3


10

0

10 Probability Density

5

10

10

Uniform Log ( 3) Log ( 11) Log ( 22) InvUnif ( 3) InvUnif ( 11) InvUnif ( 22) Posterior: Solid Prior: Dashed

10

15

10

20

Conservative
10
25

22

19

16

13 11 8 1 log10[ ] ( in Gyr )

5

3

0

3


3.5 3 2.5
(Gyr)

0

PDF
1 2

log10[Pposterior]

evolve

2 3 1.5 4 1 5 0.5 3 2

(uniform prior)
1 0 1 1 log10[ ] ( in Gyr ) 2 3 6


1 3.5 3 2.5 (Gyr) 0.6 2 1.5 1 0.2 0.5 3 2 0.1 2 3 0 0.5 0.4 0.3

CDF

0.9 0.8 0.7
0

Pposterior[ ]d

evolve

(uniform prior)
1 0 1 1 log10[ ] ( in Gyr )


3.5 3 2.5 (Gyr) 2

0

PDF
1 2

log10[Pposterior]

evolve

3 1.5 4 1 5 0.5 3 2

(logarithmic prior)
1 0 1 1 log10[ ] ( in Gyr ) 2 3 6


1 3.5 3 2.5 (Gyr) 0.6 2 1.5 1 0.2 0.5 3 2 0.1 2 3 0 0.5 0.4 0.3

CDF

0.9 0.8 0.7
0

Pposterior[ ]d

evolve

(logarithmic prior)
1 0 1 1 log10[ ] ( in Gyr )


3.5 3 2.5 (Gyr) 2

0

PDF
1 2

log10[Pposterior]

evolve

3 1.5 4 1 5 0.5 3 2

(inverse uniform prior)
1 0 1 1 log10[ ] ( in Gyr ) 2 3 6


1 3.5 3 2.5 (Gyr) 0.6 2 1.5 1 0.2 0.5 3 2 0.1 2 3 0 0.5 0.4 0.3

CDF

0.9 0.8 0.7
0

Pposterior[ ]d

evolve

(inverse uniform prior)
1 0 1 1 log10[ ] ( in Gyr )


8 6 Bounds on log10[ ] ( in Gyr 1) 4 2 0 2 4 6 8 10 10 5 0 5 1 log10[ max] ( in Gyr ) 10 15 Median Value of 1 2 Lower Bound Lower Bound

Optimistic
( min = 10 11 Gyr 1)