"Anthropic selection bias" just filters out observations that aren't compatible with our evidence. The idea that "anthropic selection bias" somehow equalizes the probability of any models which explain the evidence is provably wrong. Just wrong. (There are legitimate uses of anthropic selection bias effects, but they come up in exotic scenarios such as simulations.)
If you start from the perspective of an ideal bayesian reasoner - ala Solomonoff, you only consider theories/models that are compatible with your observations anyway.
So there are models where abiogenesis is 'easy' (which is really too vague - so let's define that as a high transition probability per unit time, over a wide range of planetary parameters.)
There are also models where abiogenesis is 'hard' - low probability per unit time, and generally more 'sparse' over the range of planetary parameters.
By Baye's Rule, we have: P(H|E) = P(E|H)P(H) / P(E)
We are comparing two hypothesises, H1, and H2, so we can ignore P(E) - the prior of the evidence, and we have:
P(H1|E) )= P(E|H1) P(H1)
P(H2|E) )= P(E|H2) P(H2)
)= here means 'proportional'
Assume for argument's sake that the model priors are the same. The posterior then just depends on the likelihood - P(E|H1) - the probability of observing the evidence, given that the hypothesis is true.
By definition, the model which predicts abiogenesis is rare has a lower likelihood.
One way of thinking about this: Abiogenesis could be rare or common. There are entire sets of universes where it is rare, and entire sets of universes where it is common. Absent any other specific evidence, it is obviously more likely that we live in a universe where it is more common, as those regions of the multiverse have more total observers like us.
Now it could be that abiogenesis is rare, but reaching that conclusion would require integrating evidence from more than earth - enough to overcome the low initial probability of rarity.
The posterior then just depends on the likelihood - P(E|H1) - the probability of observing the evidence, given that the hypothesis is true. By definition, the model which predicts abiogenesis is rare has a lower likelihood.
We are in a vast, seemingly-empty universe. Models which predict the universe should be full of life should be penalised with a lower likelihood.
...Abiogenesis could be rare or common ... it is obviously more likely that we live in a universe where it is more common, as those regions of the multiverse have more total observers like us
After a 6+ month hiatus driven by grad school and personal projects, I am finally able to continue my sequence on astrobiology. I was flabbergasted by the positive response my last post got, and despite my status as a biologist with a hobby rather than an astronomer I decided to take a more rigorously mathematical approach to figuring out our biosphere's position in space and time rather than talking in generalizations and impressions.
Post is here: http://thegreatatuin.blogspot.com/2016/03/space-and-time-revisited.html. Seeing as this post is an elaboration on the last one, I am posting a link rather than reproducing the text.
To summarize, I found some actual rigorous observational fits to the star formation rate in the universe over time and projected them into the future. These fits show the Sun as forming after 79% of all stars that will ever exist, and that 90% of all stars that will ever exist already exist. This makes sense in the light of recent work on 'galaxy quenching' - a process by which galaxies more or less completely shut off star formation through a number of processes - indicating that the majority of gas in the universe probably won't form stars if trends that have held for most of the history of the universe continue to hold. It relies heavily on analysis I began in comments on this site a few months ago.
I then lift two distinct metallicity normalizations from a paper that was making the rounds here a while back ("On The History and Future of Cosmic Planet Formation"), in an attempt to deal with the fact that that is a measurement of STAR formation, not terrestrial-planet-with-a-biosphere formation. Depending on which metallicity normalization you use (and how willing you are to take a couple naive assumptions I make in order to slot the math that is too complicated for me to comment on on top of my star formation numbers) the Earth shows up as forming after either 72% or 51% of all terrestrial planets.
These numbers are remarkable in how boring they are. We find ourselves in an utterly typical position in planet-order, even if I am wrong by quite a bit. We are not early. Of interest to many here, explanations of the so called Fermi paradox must go elsewhere, into the genesis of intelligent systems being exceedingly rare or the genesis of intelligent systems not implying interstellar spread.
Now that I seem to have a life again, I will be getting back to my original plan next, talking about our own solar system.