Life arose on Earth sometime in the first few hundred million years after the young planet had cooled to the point that it could support water-based organisms on its surface. The early emergence of life on Earth has been taken as evidence that the probability of abiogenesis is high, if starting from young-Earth-like conditions. We revisit this argument quantitatively in a Bayesian statistical framework. By constructing a simple model of the probability of abiogenesis, we calculate a Bayesian estimate of its posterior probability, given the data that life emerged fairly early in Earth's history and that, billions of years later, curious creatures noted this fact and considered its implications. We find that, given only this very limited empirical information, the choice of Bayesian prior for the abiogenesis probability parameter has a dominant influence on the computed posterior probability. Although terrestrial life's early emergence provides evidence that life might be common in the Universe if early-Earth-like conditions are, the evidence is inconclusive and indeed is consistent with an arbitrarily low intrinsic probability of abiogenesis for plausible uninformative priors. Finding a single case of life arising independently of our lineage (on Earth, elsewhere in the Solar System, or on an extrasolar planet) would provide much stronger evidence that abiogenesis is not extremely rare in the Universe.
Interesting, but I was more thinking in the "simple life" => "intelligent life" step than about abiogenesis. As you said, we don't have any strong evidence about the likelihood of abiogenesis, we only have two weak ones : the fact it happened early on Earth, and the Miller's experiment which shows that under right conditions, build blocks of life can arise quickly.
But for the "simple life" => "intelligent life" step, we have even much less evidence. We know it took a long time on Earth, about 4 billions of year...
Not entirely relevant to the main issues of lesswrong, but possibly still of interest: my talk entitled "von Neumann probes and Dyson spheres: what exploratory engineering can tell us about the Fermi paradox".
Abstract: The Fermi paradox is the contrast between the high estimate of the likelihood of extraterritorial civilizations, and the lack of visible evidence of them. But what sort of evidence should we expect to see? This is what exploratory engineering can tell us, giving us estimates of what kind of cosmic structures are plausibly constructable by advanced civilizations, and what traces they would leave. Based on our current knowledge, it seems that it would be easy for such a civilization to rapidly occupy vast swathes of the universe in a visible fashion. There are game-theoretic reasons to suppose that they would do so. This leads to a worsening of the Fermi paradox, reducing the likelihood of "advanced but unseen" civilizations, even in other galaxies.
The slides from the talk can be found here (thanks, Luke!).