4 research outputs found
Reversible Markov chain estimation using convex-concave programming
We present a convex-concave reformulation of the reversible Markov chain
estimation problem and outline an efficient numerical scheme for the solution
of the resulting problem based on a primal-dual interior point method for
monotone variational inequalities. Extensions to situations in which
information about the stationary vector is available can also be solved via the
convex- concave reformulation. The method can be generalized and applied to the
discrete transition matrix reweighting analysis method to perform inference
from independent chains with specified couplings between the stationary
probabilities. The proposed approach offers a significant speed-up compared to
a fixed-point iteration for a number of relevant applications.Comment: 17pages, 2 figure
Information Geometry of Reversible Markov Chains
We analyze the information geometric structure of time reversibility for
parametric families of irreducible transition kernels of Markov chains. We
define and characterize reversible exponential families of Markov kernels, and
show that irreducible and reversible Markov kernels form both a mixture family
and, perhaps surprisingly, an exponential family in the set of all stochastic
kernels. We propose a parametrization of the entire manifold of reversible
kernels, and inspect reversible geodesics. We define information projections
onto the reversible manifold, and derive closed-form expressions for the
e-projection and m-projection, along with Pythagorean identities with respect
to information divergence, leading to some new notion of reversiblization of
Markov kernels. We show the family of edge measures pertaining to irreducible
and reversible kernels also forms an exponential family among distributions
over pairs. We further explore geometric properties of the reversible family,
by comparing them with other remarkable families of stochastic matrices.
Finally, we show that reversible kernels are, in a sense we define, the minimal
exponential family generated by the m-family of symmetric kernels, and the
smallest mixture family that comprises the e-family of memoryless kernels
Event-chain Monte Carlo: foundations, applications, and prospects
This review treats the mathematical and algorithmic foundations of
non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC),
a continuous-time lifted Markov chain that employs the factorized Metropolis
algorithm. It analyzes a number of model applications, and then reviews the
formulation as well as the performance of ECMC in key models in statistical
physics. Finally, the review reports on an ongoing initiative to apply the
method to the sampling problem in molecular simulation, that is, to real-world
models of peptides, proteins, and polymers in aqueous solution.Comment: 35 pages, no figure