  Probabilities in eternal inflation are traditionally defined as limiting
frequency distributions, but a unique and unambiguous probability measure
remains elusive. In this paper, we present a different approach, based on
Bayesian reasoning. Our starting point is the master equation governing vacuum
dynamics, which describes a random walk on the network of vacua. Our
probabilities require two pieces of prior information, both pertaining to
initial conditions: a prior density $\rho(t)$ for the time of nucleation, and a
prior probability $p_\alpha$ for the ancestral vacuum. For ancestral vacua, we
advocate the uniform prior as a conservative choice, though our conclusions are
fairly insensitive to this choice. For the time of nucleation, we argue that a
uniform prior is consistent with the time-translational invariance of the
master equation and represents the minimally-informative choice. The resulting
predictive probabilities coincide with Bousso's "holographic" prior
probabilities and are closely related to Garriga and Vilenkin's "comoving"
probabilities. Despite making the least informative priors, these probabilities
are surprisingly predictive. They favor vacua whose surrounding landscape
topography is that of a deep funnel, akin to the folding funnels of
naturally-occurring proteins. They predict that we exist during the approach to
near-equilibrium, much earlier than the mixing time for the landscape. We also
consider a volume-weighted $\rho(t)$, which amounts to weighing vacua by
physical volume. The predictive probabilities in this case coincide with the
GSVW measure. The Bayesian framework allows us to compare the plausibility of
the uniform-time and volume-weighted hypotheses to explain our data by
computing the Bayesian evidence for each. We argue, under general and plausible
assumptions, that posterior odds overwhelmingly favor the uniform-time
hypothesis.
