Statistics of One: What Earth can and can’t tell us about life in the universe.
Presented at Bioastronomy 2004: Habitable Worlds, Reykjavik, Iceland July 1216, 2004
abstract: Astrobiology 4, 266 (2004)
Eric J. Korpela
Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450
korpela@ssl.berkeley.edu
ABSTRACT
Recent studies have attempted to extrapolate the abundance of life in the universe
and the properties of such life from the history of life on Earth. A common “reasonable” assumption in such studies is that the history of life on earth is typical of lifein the universe.(1) In this poster we examine the question of whether life on Earth islikely to be typical of life in the universe. In determining this we must consider that,as observers, we are biased by having only one example of life: the example that led
to our evolution on a 4.5 billion year old earth. This “anthropic principle,” is moreaccurately described as an anthropic bias, which alters the relationship between measured quanities such as evolutionary timescales and the distribution of values thesequantities may have in reality. In some instances this bias can be quantified. We apply
a statistical approach to quantifying this bias in the case of the timescales involved inabiogenesis and evolution of intelligent species.

Methodology
For simplicity, we model the evolution of intelligent life as two Poissonian processes, abiogenesis, and evolution of intelligence with typical timescales of τbio and τevo . In practice, thismethod works for arbitrary numbers of variables and non-Poissonian processes. We also tbio andtevo as the actual time required for abiogenesis and evolution on a specific world. Our task is, givena single measurement of tbio and tevo , what is the distribution of likely values of τbio and τevo .
is:For a given τbio , the probability of life existing on a planet identical to earth at some time t0¶µZ t0−t1dt
exp
PL (t0 ) =τbio 0τbio

–2–and the expectation value of tbio on worlds where life has arisen prior to time t0 is³ ´
τbio + (−t0 − τbio ) exp τ−tbio0³ ´
htbio i =1 − exp τ−tbio0
The probability of an intelligent species evolving prior to t0 is then¶µZ−tPL (t0 ) t0 −htbio idtexp
Pi (t0 ) =τevo 0τevo
and the expectation value of tevo on worlds where intelligence has evolved is³´bio iτevo + (htbio i − t0 − τevo ) exp −t0τ+htevo³´
htevo i =−t0 +htbio i
1 − exp
τevo
For given values of τbio and τevo such values are easily computed. However, in the real world,
τbio and τevo are the unknowns and htbio i and htbio i can only be determined from a large numberof measurements. In the case of Earth, we have only a single measurement. The process is also
complicated by the fact that the question cannot be asked until after the point t0 > (tbio + tevo ).
(Given the relative time scales of evolution and social change, one might easily assume the contraint t0 = (tbio + tevo ). We will not, however, make this assumption in our further calculations.)
In order to condsider how measurements of tbio and tevo constrain τbio and τevo we need toconsider the distribution of possible values of tbio and tevo for a prior distribution of τbio and τevo .

Likelyhood Distributions
For an unknown timescale, the appropriate unitless prior distribution representing a total lackof knowledge is:dτ= d log (τ)
P0 (τ) =τIn other words, the probable values of a timescale are equally distributed in logarithmic space.
At each possible value of τ we can determine the normalized distribution of values of measured
values of Dτ (t). The relative likelyhood of a measurement t corresponding to a value of τ isthen Lτ (t, τ) = Dτ (t) P0 (τ). Using our two parameter model, we have computed the likelyhood
distributions Lτbio (tb io, τbio , τevo ) and Lτevo (te vo, τbio , τevo ) for t0 = 4.5Gyr.
Figure 1 shows the likelyhood distributions. The contours represent a change of a factor oftwo. As would be expected, there is a significant dependence of t values on their corresponding

–3–

109

tbio [yr]tbio [yr]

109

108

107
107

107
108

109
τbio [yr]

1010

1011

108

109

1010
τevo [yr]10111012

109

tevo [yr]tevo [yr]

109

108

107
107

108

108

107
108

109
τbio [yr]

1010

108

109

1010
τevo [yr]1011Fig. 1.— Likelyhood distribution for measured values of tbio and tevo vs τbio and τbio . Horizontal
lines show limits to measurements of tbio and tevo on Earth.

–4–
τ values, with the expectation values of t roughly equal to τ at τ ≪ t0 and increasing to ∼ t0 /2 atτ ≫ t0 . There is also some cross dependence of tbio on τevo and tevo on τbio due to the constraint(tbio + tevo ) < t0 . Horizontal lines show limits to measurements of 25Myr < tbio < 600Myr and
600Myr < tevo < 4.5Gyr.
By integrating the distributions between these limits, we can generate a probability distribution for values of τbio and τevo . This distribution is shown in Figure 2.

1012

1011

τbio [yr]

1010

109

108

107
108

109

1010
τevo [yr]1011Fig. 2.— Likelyhood distribution of τbio and τevo .
Note that neither τbio nor τevo are well constrained by the timescales found on the earth. Whileτbio has a well defined peak-likelyhood value, the distribution does not go to zero at large valuesof τbio , but tends toward a constant value. The distribution of τevo on the other hand, does not havea well defined maximum likelyhood value, but trends toward a constant value. This distribution
can be used as the prior for subsequent observations, should any arise. The relative likelyhood
distributions for each parameter are shown in Figure 3.

–5–

1.0

Relative Likelyhood

0.8

0.6

0.4

0.2

0.0
1081091010
τevo [yr]1011

1.0Relative Likelyhood

0.8

0.6

0.4

0.20.0
107

108

109

τbio [yr]Fig. 3.— Likelyhood distributions for τbio and τevo .

1010

1011

1012

–6–Conclusions• When only a single example of a class is available, an observer must consider whether theobserved properties are indicative of a general property of a class. This can be done using
relatively straightforward statistical methods.
• In specific, we have examined the question of whether the observed timescales of abiogenesis
and evolution of intelligence on Earth can be used to constrain the corresponding timescales
on Earthlike planets in general.
• It is likely that the measured timescale of evolution on intelligence on earth greatly underestimates the typical timescale required.
• The early origin of life on earth results in the a well defined peak in the distribution ofpotential values of the general timescale. However, the distribution tends toward a constantvalue for large timescales. That indicates a very significant probability that the time required
for abiogenesis on earth was atypically short.
• Without further examples, other derivations, or analytical reasons restrict long duration
timescales, no upper limit can be placed on the timescales required for abiogenesis andevolution of intelligence.

REFERENCES
Lineweaver, C. H. and Davis, T. M., “Does the Rapid Appearance of Life on Earth Suggest that
Life Is Common in the Universe?,” Astrobiology 2, 293–304 (Aug. 2002).

This preprint was prepared with the AAS LATEX macros v5.2.