24 research outputs found
Fidelity of Hyperbolic Space for Bayesian Phylogenetic Inference
Bayesian inference for phylogenetics is a gold standard for computing
distributions of phylogenies. It faces the challenging problem of. moving
throughout the high-dimensional space of trees. However, hyperbolic space
offers a low dimensional representation of tree-like data. In this paper, we
embed genomic sequences into hyperbolic space and perform hyperbolic Markov
Chain Monte Carlo for Bayesian inference. The posterior probability is computed
by decoding a neighbour joining tree from proposed embedding locations. We
empirically demonstrate the fidelity of this method on eight data sets. The
sampled posterior distribution recovers the splits and branch lengths to a high
degree. We investigated the effects of curvature and embedding dimension on the
Markov Chain's performance. Finally, we discuss the prospects for adapting this
method to navigate tree space with gradients
Efficient Sampling from Feasible Sets of SDPs and Volume Approximation
We present algorithmic, complexity, and implementation results on the problem
of sampling points from a spectrahedron, that is the feasible region of a
semidefinite program. Our main tool is geometric random walks. We analyze the
arithmetic and bit complexity of certain primitive geometric operations that
are based on the algebraic properties of spectrahedra and the polynomial
eigenvalue problem. This study leads to the implementation of a broad
collection of random walks for sampling from spectrahedra that experimentally
show faster mixing times than methods currently employed either in theoretical
studies or in applications, including the popular family of Hit-and-Run walks.
The different random walks offer a variety of advantages , thus allowing us to
efficiently sample from general probability distributions, for example the
family of log-concave distributions which arise in numerous applications. We
focus on two major applications of independent interest: (i) approximate the
volume of a spectrahedron, and (ii) compute the expectation of functions coming
from robust optimal control. We exploit efficient linear algebra algorithms and
implementations to address the aforemen-tioned computations in very high
dimension. In particular, we provide a C++ open source implementation of our
methods that scales efficiently, for the first time, up to dimension 200. We
illustrate its efficiency on various data sets
Improving Gradient-guided Nested Sampling for Posterior Inference
We present a performant, general-purpose gradient-guided nested sampling
algorithm, , combining the state of the art in differentiable
programming, Hamiltonian slice sampling, clustering, mode separation, dynamic
nested sampling, and parallelization. This unique combination allows to scale well with dimensionality and perform competitively on a variety
of synthetic and real-world problems. We also show the potential of combining
nested sampling with generative flow networks to obtain large amounts of
high-quality samples from the posterior distribution. This combination leads to
faster mode discovery and more accurate estimates of the partition function.Comment: 10 pages, 5 figures. Code available at
https://github.com/Pablo-Lemos/GGN