Continuity of the Maximum-Entropy Inference

We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non-commutative observables and that a geodesic closure of a Gibbsian family equals the set of maximum-entropy states. We discuss eight descriptions of the set of maximum-entropy states with proofs of accuracy and an analysis of deviations.Comment: 34 pages, 1 figur

Information-geometric Markov Chain Monte Carlo methods using Diffusions

Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond Statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for Statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.Comment: 22 pages, 2 figure

Statistical Methods in Topological Data Analysis for Complex, High-Dimensional Data

The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data spaces. This paper provides an introductory overview of the mathematical underpinnings of Topological Data Analysis, the workflow to convert samples of data to topological summary statistics, and some of the statistical methods developed for performing inference on these topological summary statistics. The intention of this non-technical overview is to motivate statisticians who are interested in learning more about the subject.Comment: 15 pages, 7 Figures, 27th Annual Conference on Applied Statistics in Agricultur

Information Geometry of Complex Hamiltonians and Exceptional Points

Information geometry provides a tool to systematically investigate the parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric

Entropic Dynamics

Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In this paper we review three examples of entropic dynamics. First we tackle the simpler case of a standard diffusion process which allows us to address the central issue of the nature of time. Then we show that imposing the additional constraint that the dynamics be non-dissipative leads to Hamiltonian dynamics. Finally, considerations from information geometry naturally lead to the type of Hamiltonian that describes quantum theory.Comment: Invited contribution to the Entropy special volume on Dynamical Equations and Causal Structures from Observation

Latent tree models

Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in: phylogenetic analysis, network tomography, computer vision, causal modeling, and data clustering. They also contain other well-known classes of models like hidden Markov models, Brownian motion tree model, the Ising model on a tree, and many popular models used in phylogenetics. This article offers a concise introduction to the theory of latent tree models. We emphasise the role of tree metrics in the structural description of this model class, in designing learning algorithms, and in understanding fundamental limits of what and when can be learned

A Covariant Approach to Entropic Dynamics

Entropic Dynamics (ED) is a framework for constructing dynamical theories of inference using the tools of inductive reasoning. A central feature of the ED framework is the special focus placed on time. In previous work a global entropic time was used to derive a quantum theory of relativistic scalar fields. This theory, however, suffered from a lack of explicit or manifest Lorentz symmetry. In this paper we explore an alternative formulation in which the relativistic aspects of the theory are manifest. The approach we pursue here is inspired by the works of Dirac, Kuchar, and Teitelboim in their development of covariant Hamiltonian methods. The key ingredient here is the adoption of a local notion of time, which we call entropic time. This construction allows the expression of arbitrary notion of simultaneity, in accord with relativity. In order to ensure, however, that this local time dynamics is compatible with the background spacetime we must impose a set of Poisson bracket constraints; these constraints themselves result from requiring the dynamcics to be path independent, in the sense of Teitelboim and Kuchar.Comment: An extended version of work presented at MaxEnt 2016, the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; July 10-15 2016, Ghent, Belgiu