28,156 research outputs found
Level-Rank Duality in Chern-Simons Theory from a Non-Supersymmetric Brane Configuration
We derive level-rank duality in pure Chern-Simons gauge theories from a
non-supersymmetric Seiberg duality by using a non-supersymmetric brane
configuration in type IIB string theory. The brane configuration consists of
fivebranes, N D3 antibranes and an O3 plane. By swapping the fivebranes we
derive a 3d non-supersymmetric Seiberg duality. After level shifts from loop
effects, this identifies the IR of Sp(2N)_{2k-2N+2} and Sp(2k-2N+2)_{-2N} pure
Chern-Simons theories, which is a level-rank pair. We also derive level-rank
duality in a Chern-Simons theory based on a unitary group.Comment: 10 pages, 3 figure
Rank n swapping algebra for PGLn Fock--Goncharov X moduli space
The rank swapping multifraction algebra is a field of cross ratios up to -determinant relations equipped with a Poisson bracket, called the {\em swapping bracket}, defined on the set of ordered pairs of points of a circle using linking numbers. Let be a disk with points on its boundary. The moduli space is the building block of the Fock--Goncharov moduli space for any general surface. Given any ideal triangulation of , we find an injective Poisson algebra homomorphism from the rank Fock--Goncharov algebra for to the rank swapping multifraction algebra with respect to the Atiyah--Bott--Goldman Poisson bracket and the swapping bracket. Two such injective Poisson algebra homomorphisms related to two ideal triangulations and are compatible with each other under the flips
Parameter identifiability of discrete Bayesian networks with hidden variables
Identifiability of parameters is an essential property for a statistical
model to be useful in most settings. However, establishing parameter
identifiability for Bayesian networks with hidden variables remains
challenging. In the context of finite state spaces, we give algebraic arguments
establishing identifiability of some special models on small DAGs. We also
establish that, for fixed state spaces, generic identifiability of parameters
depends only on the Markov equivalence class of the DAG. To illustrate the use
of these results, we investigate identifiability for all binary Bayesian
networks with up to five variables, one of which is hidden and parental to all
observable ones. Surprisingly, some of these models have parameterizations that
are generically 4-to-one, and not 2-to-one as label swapping of the hidden
states would suggest. This leads to interesting difficulties in interpreting
causal effects.Comment: 23 page
Identifiability of parameters in latent structure models with many observed variables
While hidden class models of various types arise in many statistical
applications, it is often difficult to establish the identifiability of their
parameters. Focusing on models in which there is some structure of independence
of some of the observed variables conditioned on hidden ones, we demonstrate a
general approach for establishing identifiability utilizing algebraic
arguments. A theorem of J. Kruskal for a simple latent-class model with finite
state space lies at the core of our results, though we apply it to a diverse
set of models. These include mixtures of both finite and nonparametric product
distributions, hidden Markov models and random graph mixture models, and lead
to a number of new results and improvements to old ones. In the parametric
setting, this approach indicates that for such models, the classical definition
of identifiability is typically too strong. Instead generic identifiability
holds, which implies that the set of nonidentifiable parameters has measure
zero, so that parameter inference is still meaningful. In particular, this
sheds light on the properties of finite mixtures of Bernoulli products, which
have been used for decades despite being known to have nonidentifiable
parameters. In the nonparametric setting, we again obtain identifiability only
when certain restrictions are placed on the distributions that are mixed, but
we explicitly describe the conditions.Comment: Published in at http://dx.doi.org/10.1214/09-AOS689 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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