  Defining the energy function as the negative logarithm of the density, we
explore the energy landscape of a distribution via the tree of sublevel sets of
its energy. This tree represents the hierarchy among the connected components
of the sublevel sets. We propose ways to annotate the tree so that it provides
information on both topological and statistical aspects of the distribution,
such as the local energy minima (local modes), their local domains and volumes,
and the barriers between them. We develop a computational method to estimate
the tree and reconstruct the energy landscape from Monte Carlo samples
simulated at a wide energy range of a distribution. This method can be applied
to any arbitrary distribution on a space with defined connectedness. We test
the method on multimodal distributions and posterior distributions to show that
our estimated trees are accurate compared to theoretical values. When used to
perform Bayesian inference of DNA sequence segmentation, this approach reveals
much more information than the standard approach based on marginal posterior
distributions.
