Algebraic Methods of Classifying Directed Graphical Models

Directed acyclic graphical models (DAGs) are often used to describe common structural properties in a family of probability distributions. This paper addresses the question of classifying DAGs up to an isomorphism. By considering Gaussian densities, the question reduces to verifying equality of certain algebraic varieties. A question of computing equations for these varieties has been previously raised in the literature. Here it is shown that the most natural method adds spurious components with singular principal minors, proving a conjecture of Sullivant. This characterization is used to establish an algebraic criterion for isomorphism, and to provide a randomized algorithm for checking that criterion. Results are applied to produce a list of the isomorphism classes of tree models on 4,5, and 6 nodes. Finally, some evidence is provided to show that projectivized DAG varieties contain useful information in the sense that their relative embedding is closely related to efficient inference

Tropical Geometry of Statistical Models

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. The question addressed here is how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. A key role is played by the Newton polytope of a statistical model. Our results are applied to the hidden Markov model and to the general Markov model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion paper, "Parametric Inference for Biological Sequence Analysis

Graphical Presentations of Symmetric Monoidal Closed Theories

We define a notion of symmetric monoidal closed (SMC) theory, consisting of a SMC signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.Comment: Uses Paul Taylor's diagram

Tube algebras, excitations statistics and compactification in gauge models of topological phases

We consider lattice Hamiltonian realizations of ($d$+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalization of this strategy that is valid in any dimensions. We then apply the tube algebra approach to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an $R$-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is $n$-times compactified can be expressed in terms of another model in $n$-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.Comment: 71 page