Where is everybody?
โ Enrico Fermi
The omnipresence of uncertainty is part of why making predictions and decisions is so hard. We at Lumina advocate treatingย uncertaintyย explicitlyย in our models using probability distributions. Sadly this is not yet as common as it should be.ย ย A recent paper โDissolving the Fermi Paradoxโ (2018) is a powerful illustration of how including uncertainty can transform conclusions on the fascinating question of whether our Earth is the only place in the Universe harboring intelligent life.ย The authors, Anders Sandberg, Eric Drexler and Toby Ord (whom we shall refer to asย SDO), show elegantly that the apparent paradox is simply the result of the mistake of ignoring uncertainty, what Sam L. Savage calls the Flaw of Averages. In this article, we review their article and embed aย liveย Analyticaย version of their modelย that you can explore.
The Fermi Paradox
One day in 1950, Enrico Fermi, the Nobel prize-winning builder of the first nuclear reactor, was having lunch with a few friends in Los Alamos. They were looking at a New Yorker cartoon of cheerful aliens emerging from a flying saucer. Fermi famously asked โWhere is everybody?โ. Given the vast number of stars in the Milky Way Galaxy and the likely development of life and extraterrestrial intelligence, how come no ETs have come to visit or at least been detected? This question came to be called the โFermi Paradoxโ. Ever since, it has bothered those interested in the existence of extraterrestrial intelligence and whether we are alone in the Universe.
The Flaw of Averages on steroids
Sam L. Savage,ย in his book,ย The Flaw of Averages,ย shows how ignoring uncertainty and just working with a single mean or โmost likelyโ value for each uncertain quantity can lead to misleading results. To illustrate how dramatically this approach can distort your conclusions, SDO offer a toy example. Suppose there are nine factors that multiplied together give the probability of extraterrestrial intelligence (ETI) arising on any given star. If you use a point estimate of 0.1 for each factors, you could infer that there is aย 10^{-9}10โ9ย probability of any given star harboring ETI. There are aboutย 10^{11}1011ย stars in the Milky Way, so the probability that no star other than our own has a planet harboring intelligent life would be extremely small,ย (1-10^{-9})^{100B} โ 3.7\times 10^{-44}(1โ10โ9)100Bโ3.7ร10โ44.ย On the other hand, suppose that, based on what we know, each factor could be anywhere between 0 and 0.2, and assign aย uniform uncertaintyย over this interval, using theย probability distribution,ย Uniform(0, 0.2).
ย If you combine these distributions probabilistically, using Monte Carlo simulationย for example, the mean of the result is 0.21โover 5,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times more likely!ย
The Drake equation
Frank Drake, a radio astronomer who worked on the search for extraterrestrial intelligence (SETI), tried to formalize Fermiโs estimate of the number of ETIs. He suggested that we can estimate N, the number of detectable, intelligent civilizations in the Milky Way galaxy from what is now called the โDrake equationโ. It is sometimes referred to as the โsecond most-famous equation in science (after E= mc2)โ:
N= R^* \times f_p \times n_e \times f_l \times f_i \times f_c \times LN=RโรfpรneรflรfiรfcรL
Where
R^*Rโย is the average rate of formation of stars in our galaxy,
f_pfpย is the fraction of stars with planets.
n_eneย is the average number of those planets that could potentially support life.
f_lflย is the fraction of those on which life had actually developed;
f_ifiย is the fraction of those with life that is intelligent; and
f_cfcย is the fraction of those that have produced a technology detectable to us.
LLย is the average lifetime of such civilizations
Many have tried to refine this calculation since Drake first proposed it in 1961. Most have estimated a large number for N, the number of detectable extraterrestrial civilizations. The contradiction between expected proliferation of detectable ETs and their apparent absence is what came to be called the โFermi paradoxโ after the famous lunch conversation.
Past explanations of the Fermi Paradox
Many have tried to resolve the apparent paradox: Maybe advanced civilizations avoid wasteful emission of electromagnetic radiation into space that we could detect. Maybe interstellar travel is simply impossible. Or if it is technically possible, all ETs have decided itโs not worth the effort. Or perhaps ETs do visit us but choose to be discreet, deeming us not ready for the shock of contact. Maybe there is a Great Filter that makes the progression of life to advanced stages exceedingly rare. Or perhaps, the development of life from lifeless chemicals (abiogenesis) and/or the evolution of technological intelligence are just so unlikely that we are in fact the only ones in the Galaxy. Or, even more depressingly,ย perhapsย those intelligent civilizations that do emerge all manage to destroy themselves in short order before perfecting interstellar communicationโas indeed we Earthlings may plausibly do ourselves.
Quantifying uncertainty in the Drake equation
SDO propose a nice way to resolve the apparent paradox without resorting toย anyย speculative explanation. Recognizing that most of the factors of the Drake equation are highly uncertain, they express each factor as a probability distribution that characterizes the uncertainty based on their review of the relevant scientific literature. They then use simpleย Monte Carlo simulationย to estimate the probability distribution on N, and hence the probability thatย N<1N<1ย โ i.e. that there are zero ETIs to detect.ย They estimate this probability at about 52% (our reimplementation of their model comes up with 48%).ย In other words, we shouldnโt be surprised at our failure to observe any ETI because there is a decent probability that there arenโt any. Thus, we may view the Fermi paradox as due simply to Sam Savageโs โFlaw of Averagesโ: If you use only โbest estimatesโ and ignore the range of uncertainty in each assumption, youโll end up with a misleading result.
For most factors. the uncertainty ranges over many orders of magnitude. For all except one factor, SDO represent its uncertainty using aย Log-Uniform distribution, assuming that each order of magnitude is equally likely over its range.ย In other words, the logarithm of the value is uniformly-distributed. This table summarizes their estimated uncertainty for each factor.
Factor | Distribution | ย | Description |
---|---|---|---|
R^*Rโ | LogUniform(1, 100) |
Rate of star formation (stars/year) | |
f_pfp | LogUniform(0.1, 1) |
Fraction of stars with planets | |
n_ene | LogUniform(0.1, 1) |
Number ofย habitable planetary objects per system with planets (planets/star) | |
f_lfl | Version 1: LogNormalโ1-e^{-e^{m}}1โeโem where m~ Normal(0,50) |
Version 2:โt V \lambdatVฮปย versionโ1-e^{-t V \lambda}1โeโtVฮปt\simtโผย LogUniform(1e7, 1e10) V\simVโผย LogUniform(1e2, 1e15) \lambda\simฮปโผย LogUniform(1e-188, 1e15) ย |
Fraction of habitable planets that develop life.ย Abiogenesisย refers to the formation of life out of inanimate substances.t\simtโผย Time avail. for abiogenesis (years)V\simVโผย Volume of substrate for abiogenesis (m^3m3)\lambda\simฮปโผย Rate of abiogenesis (events perย m^3m3ย years)The scientific notationย 1e15 ย is a way of writingย 10^{15}1015, and so on. |
f_ifi | LogUniform(0.001, 1) |
Fraction of planets with life that develops intelligence | |
f_cfc | LogUniform(0.01, 1) |
Fraction of intelligent civilizations that are detectable | |
LL | LogUniform(100, 1e10) |
Duration of detectability (years) |
SDO present two ways to estimateย f_lfl, the fraction of habitable planets that develop life. Both use the formย 1 โ e^{-r}1โeโrย as the probability that one or more abiogenesis events occur, assuming aย Poisson-processย with rateย rr. In Version 1 they estimate aย Lognormal distributionย directly forย rr.ย Version 2 decomposesย rrย into a product of three other quantities,ย t V \lambdatVฮปย , and assigns a loguniform to each one. (It is not clear to us that these three subquantities are any easier to estimate!) At the risk of stating the obvious, the ranges forย f_lflย used by SDO are enormous in either version.
We couldnโt tell from the text of the paper alone which results used which version, so we included both versions in our model.ย This table gives some results from these models.
ย | N = # detectable planets in Milky Way | Pr(N<1) โwe are aloneโ |
Pr(N>100M) โTeeming with intelligent civilizationsโ |
|
---|---|---|---|---|
Median | Mean | |||
Reported in SDO | 0.32 | 27 million | 52% | โ |
Version 1 with uncertainty | 1.8 | 27.8 million | 48% | 1.9% |
Version 2 with uncertainty | 9.9e-67 | 8.9M | 84% | 0.6% |
Version 1 with point estimates | 2000 | 0% | 0% | |
Version 2 with point estimates | 1e-66 | 100% | 0% | |
f_l = 16%fl=16 | 500 | 8.9 million | 17% | 1.2% |
f_l\simflโผย Beta(1, 10) | 170 | 5 million | 23% | 1.4% |
The top row, โReported in SDOโ, shows the numbers from their text.ย The rest are from our Analytica implementation of their model. Their reported values seem more consistent with Version 1; but other results in their paper seem more consistent Version 2. We believe our implementation of both versions reflects those described in the paper. We even examinedย their Python codeย in a futile attempt to explain why our results arenโt an exact match. We have emailed the first author in the hope he can clarify the situation. While we canโt reproduce their exact results the discrepancies do not affect their broad qualitative conclusions.
The rows, Versions 1 and 2 with uncertainty, use SDOโs full distributions. The rows, Versions 1 and 2 with point estimates, use the median of their distributions as a point estimate for each of the seven factors of the Drake Equation or 9 parameters for Version 2.ย In Version 1 with uncertainty, the mean for N is four orders of magnitude larger than the corresponding point estimate. In Version 2 with uncertainty, it is 73 orders of magnitude larger than the corresponding point estimate.
The P(N<1) column shows the probability that there is no other detectable civilization in the Milky Way. The fact that it is so high means that we should not be surprised by Fermiโs observation that we havenโt detected any extraterrestrial civilization. In each case with uncertainty, there is a substantial probability (from 17% to 84%) that no other detectable civilization exists. We added a last column with the probability that our galaxy is absolutely teeming with lifeโ with over 100 million civilizations, or 1 out of every thousand stars having a detectable intelligent civilization. The uncertain models give us between 0.6% and 1.9% of this case.
Explore the model yourself
Our Analytica model is running here where you can explore. (or,ย click here to run it in its own browser tab).
https://acp.analytica.com/view?invite=4387&code=4370702823944257389
Here are some things to try while exploring the model.
- Click on a term in the Drake Equation for a description.
- Select a particular โmodelโ for f_l, the fraction of planets or planetary objects where life begins, or keep it at All to run all of them.
View each of the results in the UI above.
- Calculate N and view the statistics view to see the Mean and Median. Select Mid to see what the result would be without including uncertainty.
- View the PDF for LogTen(N).
- Select a observation method (or multiple ones) and see how the results change in the posterior compared to the prior.
- Click on Model Internals to explore the full implementation.
- ย
Fraction of habitable planets that develop life,ย f_lfl
The largest source of uncertainty is factorย f_lfl, the fraction of habitable planets with life.ย ย Microscopic fossils imply that life started on Earth around 3.5 to 3.8 billion years ago, quite soon after the planet formed. This suggests that abiogenesis is easy and nearly inevitable on a habitable planet. On the other hand, every known living creature on Earth uses essentially the same DNA-based genetic code, which suggests abiogenesis occurred only once in the planetโs history. So perhaps it was an astoundingly rare event that just happened to occur on Earth. The fact that it did occur here doesnโt give us information aboutย f_lflย other than the fact thatย f_lflย is not exactly zero due to anthropic biasโthe observation that we exist would be the same whether life on Earth was an incredibly rare accident or whether it was inevitable.
SDO reflect the lack of information onย f_lflย by the immense range of uncertainty for in both versions of their model. Here is their probability density function (PDF) forย f_lfl:
The PDF looks similar for Version 1 and Version 2, with spikes atย f_l\approx 0flโ0ย andย f_l\approx 1flโ1, and little probability mass between these extremes. In Version 1 the spikes are roughly equal, whereas in Version 2 the spikes atย f_l\approx 0flโ0ย andย f_l\approx 1flโ1ย have aboutย 84% and 16% probability respectively. In other words, there is a 16% chance that almost every habitable planet develops life, and an 84% probability that essentially none do. (The Earth did, of course, but this isnโt inconsistent withย f_l\approx 0flโ0ย since these values are positive, just extremely small.)ย Thus, the distribution nearly degenerates into a Bernoulli (point) probability distribution. A Bernoulli point probabilityย f_l=0.16fl=0.16ย would mean that 16% of habitable planets develop life, which is a somewhat different interpretation. To see this difference, we includedย f_l=0.16fl=0.16ย in the results as a point of comparison (See the penultimate row of the table) above.
The core problem here is that the range they used for abiogenesis events per habitable planet,ย f_lfl, just seems implausibly large in both versions, with the 25-75 quartile ranging from 2e-15 toย 4e+14. We think this may be too extreme. The nice thing about having a live model to play with is that it is possible to repeat the results using more sane alternatives.
Number of detectable civilizations
Because the model includes information about how uncertain each factor is, we can plot the probability distribution forย NN, the number of detectable civilizations in the Milky Way. Here is SDOโs distribution as a PDF:
These two are from the Analytica model for the two versions forย f_lfl.
The similarity between the first and third density, a combination of roughly LogNormal centered around Log(N)=2, and a LogUniform down toย 10^{-160}10โ160ย suggests SDO used Version 2 ofย f_lflย for this graph. However, as we mentioned, the numbers given in the text are more consistent with Version 1.
A probability density at a particular x value is obtained by estimating (by Monte Carlo simulation) the probability that the true value is within a small interval of widthย \epsilonฯตย around x, and then dividing byย \epsilonฯตย to get the density. The probability density ofย \log_{10} Nlog10Nย is not the same as the density ofย NNย since the denominator is different. Although SDO label it the graph as the probability density of N, they are actually showing the density ofย \log_{10} Nlog10N, which is a sensible scale for order-of-magnitude of uncertainty.ย They label the vertical axis as โfrequencyโ rather than probability density, likely an artifact of the binning algorithm used to estimate the densities.
Cumulative Probability Functions (CDFs)ย avoid these complications โ the y-scale is cumulative probability, whether you plotย NNย orย \log_{10} Nlog10N:
These CDFs show a dramatic difference between Version 1 (using the LogNormal method) and Version 2 (using theย t V \lambdatVฮปย method), and between those versions and ours that remove the massive lower tails. An interesting aspect of these graphs are their qualitative shape. In the PDF, they all have the familiar bell-shaped body, but the extreme left tail stands out as unusual. The previous section points out that both versions ofย f_lflย are so extreme the effective distribution is degenerate. We think this is a flaw. Hence, it is interesting to see how the graph changes when we setย f_lflย to a less degenerate distribution.
The LogNormal method forย f_lfl, and theย t V \lambdatVฮปย method are their Versions 1 and 2. The other three methods are less extreme. 100% and 16% use these as point probabilities forย f_lfl, andย Betaย uses aย Beta(1,10)
ย distribution forย f_lfl. The key conclusion of the paper that there is a significant probability that N is zero remains robust with these less extreme distributions forย f_lfl.
Bayesian updating on Fermiโs observation
Fermiโs question โwhere is everybodyโ refers to the observation that we havenโt detected any extraterrestrial civilizations. SDO apply Bayesโ rule to update the estimates with this observation. To apply Bayesโ rule, you need the likelihoodsย P(ยฌD|N)P(ยฌDโฃN) for each possible value of N, where ยฌD is the observation that no ETI has been detected. SDO explore four models for this updating.
- Random samplingย assumes that we have sampledย Kย stars, none of which harbor a detectable civilization.
- Spatial Poissonย assumes that there is no detectable civilization within a distanceย dย of Earth.
- Settlementย attemptsย to incorporate the possibility that interstellar propagation would be likely among advanced civilizations. It introduces several new parameters, including settlement timescales and a geometric factor. It is conditioned on the observation that no spacetime volume near the Earth has been permanently settled.
- No K3 observedย conditions on the observation that no Kardashev type 3 civilizations existโ civilizations that harness energy at the galactic scale. It presumes that if such a civilization exists, either in the Milky Way or even in another visible galaxy, we would have noticed it. Among other parameters, it includes one for the probability that a K3 civilization is theoretically possible.
We implemented all these update models in the Analytica model. Our match to the paperโs quantitative results is only approximate. We are not sure why the results are not precisely reproducible. It was quite challenging figuring out what parameters they used for each case, sinceย the paperย and itsย Supplement 2ย left out many details. With the exception of the settlement update model, we were able to get fairly close at least in qualitative terms. We explored that space of possible parameter values for the Settlement model but were unable to match the qualitative shape of the posterior reported in the paper. The likelihood equation for the K3 update appears to be in error in the paper since it doesnโt depend at all on N, but a more plausible version that does depend on N appears in Supplement 2.
In our Analytica model, you can select which update model(s) you want to view, and graph them side-by-side, along with (or without) the prior. For example,
Once of the more interesting posterior results isย P(N<1)P(N<1).
SDO report these numbers in Table 2 (in the same order as the rows of the above table) for P(N<1): 52%, 53%, 57%, 66%, 99.6%.ย We think they may have based their first 4 posteriors on Version 1. We are not sure about the K3 posterior, which differs substantially from our results.
In this table, we see that the models with a non-extreme, non-degenerate version ofย f_lflย are not substantially changed by the posterior update on the negative Fermi observation. These are the models that use a point estimate forย f_lflย of 100% and 16%, as well as the one that usesย f_l \sim Beta(1,10)flโผBeta(1,10).
How to compute the posteriors
We explored two ways to implement these posterior calculations in Analytica. We found their results to be consistent, so we stuck with the more interesting and flexible method. This is interesting in its own right, and very simple to code in Analytica. The calculation usesย sample weighting, in which each Monte Carlo sample is weighted byย P(ยฌD|N)P(ยฌDโฃN). The value for N is computed at each Monte Carlo sample, so from thatย P(ยฌD|N)P(ยฌDโฃN)ย is also computed for each selected posterior method. The variable that computesย P(ยฌD|N)P(ยฌDโฃN)ย has the identifierย P_obs_given_N
. To compute the posteriors, all we had to do was set the system variableย SampleWeighting
ย toย P_obs_given_N
.
We also tried a second method for computing the posterior. It gives the same results, but is less elegant and more complex. This method extracts the histogramย PDF(LogTen_N)
. It computesย P(ยฌD|N)P(ยฌDโฃN)ย based on the value of N that appears in the PDF.ย The product of the PDF column for LogTen_N and isย P(ยฌD|N)P(ยฌDโฃN)ย is the unnormalized PDF forย P(N|ยฌD)P(NโฃยฌD).
We would expect this second method to perform better than the first method when the likelihoodย P(ยฌD|N)P(ยฌDโฃN)ย is extremelyย leptokurtic. In this model, this is not the case.
Updating on a positive observation
The Fermi observation is the negative observation that we have never detected another extraterrestrial civilization. We thought it would be interesting here to explore what happens when you condition on a positive observation.
Extraterrestrial microbes
In March 2011, one of us (Lonnie Chrisman) attended a talk at Foothill Collegeย by NASA planetary scientist Dr. Chris McKay. Six years earlier, with the Huygens probe descending into the atmosphere of Saturnโs moon Titan, he and graduate student, Heather Smith, undertook a thought experiment. They asked: If there is life on Titan, what chemical signatures might we see? Especially, signatures that could not result from a known inanimate process? What would organisms eat? What would their waste products be?
At -190ยฐC (-290ยฐF), life on Titan would be very different, not based on water, but rather on liquid methane. Without knowing what such life forms would look like, they could still make some inferences about what chemical bonds organisms would likely utilize for metabolism. For example, the molecule with the most harvestable energy on Titan is acetylene (ย C_2H_2 + 3H_2 \rarr 2 CH_4, \nabla G=80 kcal/moleC2H2+3H2โ2CH4,โG=80kcal/moleย ). They published a set of proposed signatures for life in
C.P. McKay and H.D.Smith (2005), โPossibilities for methanogenic life in liquid methane on the surface of Titanโ, ICARUS 178(1):274-276, doi.org/10.1016/j.icarus.2005.05.018
and then moved on to other work. Five years later, analysis of data from the Huygenโs probe and Cassini mission to Saturn found some unexplained chemical signatures.
Darrell F.Strobel (2010), โMolecular hydrogen in Titanโs atmosphere: Implications of the measured tropospheric and thermospheric mole fractionsโ, ICARUS 208(2):878-886, doi.org/10.1016/j.icarus.2010.03.003
These signatures matched those predicted as possible signatures of life by McKay and Smith 5 years earlier. One signature, a net downward flux of hydrogen, is particularly intriguing, since it implies that something is absorbing or converting hydrogen near the surface, for which no inorganic processes are known. The data remain ambiguous. For example, the Huygens probe did not detect a depletion of hydrogen near the surface, which is what would be expected if organisms on the surface are consuming hydrogen.
The interesting question here is how we should update our uncertainty based on a hypothetical future discovery of microbial life on an extraterrestrial body such as Titan. Such a discovery would influence our belief aboutย n_ene, the number of planetary objects per star that are potentially habitable, as well asย f_lfl, the fraction of habitable planetary objects where life actually starts. In our own solar system, it would make Jupiterโs moons Europa and Enceladus more likely candidates. Adding moons to planets as possible sites for abiogenesis would increase our estimate forย n_eneย by something like a factor of 3:
P( n_e | D ) =P(neโฃD)=ย LogUniform( 0.3, 3 )
To updateย f_lfl, weโll useย P( D | f_l, habitable) = f_lP(Dโฃfl,habitable)=fl, where D is the observation that this one additional planetary object is habitable and where life has emerged.
With these updates, the probability that there is no other intelligent contactable civilization would drop from 48.5% to 6.6%, and the probability that the galaxy is teeming with intelligent life would increase from under 2% to over 6%, using the Version 1 prior. Here is the table for different version of the prior (where onlyย f_lflย varies between the 5 priors).
ย | P(N<1)ย โย โwe are aloneโ | P(N>100) โ galaxy teeming with life | ||
---|---|---|---|---|
Prior | Posterior | Prior | Posterior | |
Version 1 | 48.5% | 6.6% | 1.9% | 6.2% |
Version 2 | 84% | 6.6% | 0.6% | 6.2% |
f_l=100%fl=100 | 10% | 6.5% | 3.7% | 6.3% |
f_l=16%fl=16 | 17% | 12.7% | 1.2% | 2.5% |
f_l \simflโผBeta(1,10) |
23% | 13.6% | 1.4% | 2.4% |
It is interesting when comparing these priors how the extreme priors (Version 1 and Version 2) adjust to be nearly identical to the result obtained when settingย f_l=100%fl=100. Theย f_l=100%fl=100ย models the most extreme assumption that life always starts on every habitable planet. This reinforces our earlier criticism that the paperโs two versions ofย f_lflย are flawed. Because they are so extreme, they are roughly equivalent to saying life startsย ย on a habitable planetary bodyย either almost always or almost never. Thus the evidence that it happened on Titan would leave us with only the former option, that it happens almost always.
Summary
It is easy to be misled without realizing it when you estimate a single number (a point estimate) for an unknown quantity. Fermiโs question of why we never detected or encountered other extraterrestrial civilizations has spawned decades of conjecture for underlying reasons, yetย Sandberg, Drexler and Ordย show that there may be no paradox after all. Weโve reviewed and reimplemented the model they proposed. The possibility that there are no otherย detectableย intelligent civilizations in the Milky Way is consistent with our level of uncertainty. The apparent paradox was simply the result of the โFlaw of Averagesโ.
We hope you are able to learn something by playing with the model.ย Enjoy!