P(life is common|life on earth)=P(life is common), because knowing that life did evolve on earth can't give us Bayesian evidence for or against the hypothesis that life is common.
That math is rather obviously wrong. You are so close here - just use Bayes.
Perhaps I should have used an approximately equal to symbol instead of an equals sign, to avoid confusion. And thanks for the detailed writeup. I would agree 100% if you substituted "planet X" for "earth". Basically, I'm arguing that using ourselves as a data point is a form of the observational selection effect, just like survivorship bias.
As for the math, I'll pull an example from An Intuitive Explanation of Bayes' Theorem:
Similarly, let's suppose that we have a less discriminating test, mammography, that still has a 20% rate of false negatives, as in the original case. However, mammography has an 80% rate of false positives. In other words, a patient without breast cancer has an 80% chance of getting a false positive result on her mammography test. If we suppose the same 1% prior probability that a patient presenting herself for screening has breast cancer, what is the chance that a patient with positive mammography has cancer?
Group 1: 100 patients with breast cancer.
Group 2: 9,900 patients without breast cancer.
After mammography* screening:
Group A: 80 patients with breast cancer and a "positive" mammography*.
Group B: 20 patients with breast cancer and a "negative" mammography*.
Group C: 7920 patients without breast cancer and a "positive" mammography*.
Group D: 1980 patients without breast cancer and a "negative" mammography*.
The result works out to 80 / 8,000, or 0.01. This is exactly the same as the 1% prior probability that a patient has breast cancer! A "positive" result on mammography doesn't change the probability that a woman has breast cancer at all. You can similarly verify that a "negative" mammography also counts for nothing. And in fact it must be this way, because if mammography has an 80% hit rate for patients with breast cancer, and also an 80% rate of false positives for patients without breast cancer, then mammography is completely uncorrelated with breast cancer.
In that example, the reason the posterior probability equals the prior probability is that the "test" isn't causally linked with the cancer. You have to assume the same the same sort of thing for cases in which you are personally entangled. For example, if I watched my friend survive 100 rounds of solo Russian Roulette, then Baye's theorem would lead me to believe that there was a high probability that the gun was empty or only had 1 bullet. However, if I myself survived 100 rounds, I couldn't afterward conclude a low probability, because there would be no conceivable way for me to observe anything but 10 wins. I can't observe anything if I'm dead.
Does what I'm saying make sense? I'm not sure how else to put it. Are you arguing that Baye's theorem can still output good data even if you feed it skewed evidence? Or are you arguing that the evidence isn't actually the result of survivorship bias/observation selection effect?
For example, if I watched my friend survive 100 rounds of solo Russian Roulette, then Baye's theorem would lead me to believe that there was a high probability that the gun was empty or only had 1 bullet. However, if I myself survived 100 rounds, I couldn't afterward conclude a low probability, because there would be no conceivable way for me to observe anything but 10 wins. I can't observe anything if I'm dead.
Obviously you can't observe anything if you are dead, but that isn't interesting. What matter is comparing the various hypothesis that could ex...
Our sun appears to be a typical star: unremarkable in age, composition, galactic orbit, or even in its possession of many planets. Billions of other stars in the milky way have similar general parameters and orbits that place them in the galactic habitable zone. Extrapolations of recent expolanet surveys reveal that most stars have planets, removing yet another potential unique dimension for a great filter in the past.
According to Google, there are 20 billion earth like planets in the Galaxy.
A paradox indicates a flaw in our reasoning or our knowledge, which upon resolution, may cause some large update in our beliefs.
Ideally we could resolve this through massive multiscale monte carlo computer simulations to approximate Solonomoff Induction on our current observational data. If we survive and create superintelligence, we will probably do just that.
In the meantime, we are limited to constrained simulations, fermi estimates, and other shortcuts to approximate the ideal bayesian inference.
The Past
While there is still obvious uncertainty concerning the likelihood of the series of transitions along the path from the formation of an earth-like planet around a sol-like star up to an early tech civilization, the general direction of the recent evidence flow favours a strong Mediocrity Principle.
Here are a few highlight developments from the last few decades relating to an early filter:
The Future(s)
When modelling the future development of civilization, we must recognize that the future is a vast cloud of uncertainty compared to the past. The best approach is to focus on the most key general features of future postbiological civilizations, categorize the full space of models, and then update on our observations to determine what ranges of the parameter space are excluded and which regions remain open.
An abridged taxonomy of future civilization trajectories :
Collapse/Extinction:
Civilization is wiped out due to an existential catastrophe that sterilizes the planet sufficient enough to kill most large multicellular organisms, essentially resetting the evolutionary clock by a billion years. Given the potential dangers of nanotech/AI/nuclear weapons - and then aliens, I believe this possibility is significant - ie in the 1% to 50% range.
Biological/Mixed Civilization:
This is the old-skool sci-fi scenario. Humans or our biological descendants expand into space. AI is developed but limited to human intelligence, like CP30. No or limited uploading.
This leads eventually to slow colonization, terraforming, perhaps eventually dyson spheres etc.
This scenario is almost not worth mentioning: prior < 1%. Unfortunately SETI in current form is till predicated on a world model that assigns a high prior to these futures.
PostBiological Warm-tech AI Civilization:
This is Kurzweil/Moravec's sci-fi scenario. Humans become postbiological, merging with AI through uploading. We become a computational civilization that then spreads out some fraction of the speed of light to turn the galaxy into computronium. This particular scenario is based on the assumption that energy is a key constraint, and that civilizations are essentially stellavores which harvest the energy of stars.
One of the very few reasonable assumptions we can make about any superintelligent postbiological civilization is that higher intelligence involves increased computational efficiency. Advanced civs will upgrade into physical configurations that maximize computation capabilities given the local resources.
Thus to understand the physical form of future civs, we need to understand the physical limits of computation.
One key constraint is the Landauer Limit, which states that the erasure (or cloning) of one bit of information requires a minimum of kTln2 joules. At room temperature (293 K), this corresponds to a minimum of 0.017 eV to erase one bit. Minimum is however the keyword here, as according to the principle, the probability of the erasure succeeding is only 50% at the limit. Reliable erasure requires some multiple of the minimal expenditure - a reasonable estimate being about 100kT or 1eV as the minimum for bit erasures at today's levels of reliability.
Now, the second key consideration is that Landauer's Limit does not include the cost of interconnect, which is already now dominating the energy cost in modern computing. Just moving bits around dissipates energy.
Moore's Law is approaching its asymptotic end in a decade or so due to these hard physical energy constraints and the related miniaturization limits.
I assign a prior to the warm-tech scenario that is about the same as my estimate of the probability that the more advanced cold-tech (reversible quantum computing, described next) is impossible: < 10%.
From Warm-tech to Cold-tech
There is a way forward to vastly increased energy efficiency, but it requires reversible computing (to increase the ratio of computations per bit erasures), and full superconducting to reduce the interconnect loss down to near zero.
The path to enormously more powerful computational systems necessarily involves transitioning to very low temperatures, and the lower the better, for several key reasons:
Assuming large scale quantum computing is possible, then the ultimate computer is thus a reversible massively entangled quantum device operating at absolute zero. Unfortunately, such a device would be delicate to a degree that is hard to imagine - even a single misplaced high energy particle could cause enormous damage.
Stellar Escape Trajectories
The Great Game
If two civs both discover each other's locations around the same time, then MAD (mutually assured destruction) dynamics takeover and cooperation has stronger benefits. The vast distances involve suggest that one sided discoveries are more likely.
Spheres of Influence
Conditioning on our Observational Data
Observational Selection Effects
All advanced civs will have strong instrumental reasons to employ deep simulations to understand and model developmental trajectories for the galaxy as a whole and for civilizations in particular. A very likely consequence is the production of large numbers of simulated conscious observers, ala the Simulation Argument. Universes with the more advanced low temperature reversible/quantum computing civilizations will tend to produce many more simulated observer moments and are thus intrinsically more likely than one would otherwise expect - perhaps massively so.
Rogue Planets
Although the error range is still large, it appears that free floating planets outnumber planets bound to stars, and perhaps by a rather large margin.
Assuming the galaxy is colonized: It could be that rogue planets form naturally outside of stars and then are colonized. It could be they form around stars and then are ejected naturally (and colonized). Artificial ejection - even if true - may be a rare event. Or not. But at least a few of these options could potentially be differentiated with future observations - for example if we find an interesting discrepancy in the rogue planet distribution predicted by simulations (which obviously do not yet include aliens!) and actual observations.
Also: if rogue planets outnumber stars by a large margin, then it follows that rogue planet flybys are more common in proportion.
Conclusion
SETI to date allows us to exclude some regions of the parameter space for alien civs, but the regions excluded correspond to low prior probability models anyway, based on the postbiological perspective on the future of life. The most interesting regions of the parameter space probably involve advanced stealthy aliens in the form of small compact cold objects floating in the interstellar medium.
The upcoming WFIST telescope should shed more light on dark matter and enhance our microlensing detection abilities significantly. Sadly, it's planned launch date isn't until 2024. Space development is slow.