  We address phylogenetic reconstruction when the data is generated from a
mixture distribution. Such topics have gained considerable attention in the
biological community with the clear evidence of heterogeneity of mutation
rates. In our work, we consider data coming from a mixture of trees which share
a common topology, but differ in their edge weights (i.e., branch lengths). We
first show the pitfalls of popular methods, including maximum likelihood and
Markov chain Monte Carlo algorithms. We then determine in which evolutionary
models, reconstructing the tree topology, under a mixture distribution, is
(im)possible. We prove that every model whose transition matrices can be
parameterized by an open set of multi-linear polynomials, either has
non-identifiable mixture distributions, in which case reconstruction is
impossible in general, or there exist linear tests which identify the topology.
This duality theorem, relies on our notion of linear tests and uses ideas from
convex programming duality. Linear tests are closely related to linear
invariants, which were first introduced by Lake, and are natural from an
algebraic geometry perspective.
