  It is unclear how frequently life and intelligence arise on planets. I
consider a Bayesian prior for the probability P(ETI) that intelligence evolves
at a suitable site, with weight distributed evenly over ln(1 - ln P(ETI)). This
log log prior can handle a very wide range of P(ETI) values, from 1 to
10^(-10^122), while remaining responsive to evidence about extraterrestrial
societies. It is motivated by our uncertainty in the number of conditions that
must be fulfilled for intelligence to arise, and it is related to
considerations of information, entropy, and state space dimensionality. After
setting a lower limit to P(ETI) from the number of possible genome sequences, I
calculate a Bayesian confidence of 18% that aliens exist within the observable
Universe. With different assumptions about the minimum P(ETI) and the number of
times intelligence can appear on a planet, this value falls between 1.4% and
47%. Overall, the prior leans towards our being isolated from extraterrestrial
intelligences, but indicates that we should not be confident of this
conclusion. I discuss the implications of the prior for the Search for
Extraterrestrial Intelligence, concluding that searches for interstellar probes
from nearby societies seem relatively effective. I also discuss the possibility
of very small probabilities allowed by the prior for the origin of life and the
Fermi Paradox, and note that similar priors might be constructed for
interesting complex phenomena in general.
