3,849 research outputs found
Quantum Graphical Models and Belief Propagation
Belief Propagation algorithms acting on Graphical Models of classical
probability distributions, such as Markov Networks, Factor Graphs and Bayesian
Networks, are amongst the most powerful known methods for deriving
probabilistic inferences amongst large numbers of random variables. This paper
presents a generalization of these concepts and methods to the quantum case,
based on the idea that quantum theory can be thought of as a noncommutative,
operator-valued, generalization of classical probability theory. Some novel
characterizations of quantum conditional independence are derived, and
definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and
Bayesian Networks are proposed. The structure of Quantum Markov Networks is
investigated and some partial characterization results are obtained, along the
lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation
algorithm is presented and is shown to converge on 1-Bifactor Networks and
Markov Networks when the underlying graph is a tree. The use of Quantum Belief
Propagation as a heuristic algorithm in cases where it is not known to converge
is discussed. Applications to decoding quantum error correcting codes and to
the simulation of many-body quantum systems are described.Comment: 58 pages, 9 figure
Efficient algorithms for conditional independence inference
The topic of the paper is computer testing of (probabilistic) conditional independence (CI) implications by an algebraic method of structural imsets. The basic idea is to transform (sets of) CI statements into certain integral vectors and to verify by a computer the corresponding algebraic relation between the vectors, called the independence implication. We interpret the previous methods for computer testing of this implication from the point of view of polyhedral geometry. However, the main contribution of the paper is a new method, based on linear programming (LP). The new method overcomes the limitation of former methods to the number of involved variables. We recall/describe the theoretical basis for all four methods involved in our computational experiments, whose aim was to compare the efficiency of the algorithms. The experiments show that the LP method is clearly the fastest one. As an example of possible application of such algorithms we show that testing inclusion of Bayesian network structures or whether a CI statement is encoded in an acyclic directed graph can be done by the algebraic method
Total positivity in exponential families with application to binary variables
We study exponential families of distributions that are multivariate totally
positive of order 2 (MTP2), show that these are convex exponential families,
and derive conditions for existence of the MLE. Quadratic exponential familes
of MTP2 distributions contain attractive Gaussian graphical models and
ferromagnetic Ising models as special examples. We show that these are defined
by intersecting the space of canonical parameters with a polyhedral cone whose
faces correspond to conditional independence relations. Hence MTP2 serves as an
implicit regularizer for quadratic exponential families and leads to sparsity
in the estimated graphical model. We prove that the maximum likelihood
estimator (MLE) in an MTP2 binary exponential family exists if and only if both
of the sign patterns and are represented in the sample for
every pair of variables; in particular, this implies that the MLE may exist
with observations, in stark contrast to unrestricted binary exponential
families where observations are required. Finally, we provide a novel and
globally convergent algorithm for computing the MLE for MTP2 Ising models
similar to iterative proportional scaling and apply it to the analysis of data
from two psychological disorders
Geometry of rank tests
We study partitions of the symmetric group which have desirable geometric
properties. The statistical tests defined by such partitions involve counting
all permutations in the equivalence classes. These permutations are the linear
extensions of partially ordered sets specified by the data. Our methods refine
rank tests of non-parametric statistics, such as the sign test and the runs
test, and are useful for the exploratory analysis of ordinal data. Convex rank
tests correspond to probabilistic conditional independence structures known as
semi-graphoids. Submodular rank tests are classified by the faces of the cone
of submodular functions, or by Minkowski summands of the permutohedron. We
enumerate all small instances of such rank tests. Graphical tests correspond to
both graphical models and to graph associahedra, and they have excellent
statistical and algorithmic properties.Comment: 8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/.
v2: Expanded proofs, revised after reviewer comment
Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?
Correlations in the signal observed via functional Magnetic Resonance Imaging
(fMRI), are expected to reveal the interactions in the underlying neural
populations through hemodynamic response. In particular, they highlight
distributed set of mutually correlated regions that correspond to brain
networks related to different cognitive functions. Yet graph-theoretical
studies of neural connections give a different picture: that of a highly
integrated system with small-world properties: local clustering but with short
pathways across the complete structure. We examine the conditional independence
properties of the fMRI signal, i.e. its Markov structure, to find realistic
assumptions on the connectivity structure that are required to explain the
observed functional connectivity. In particular we seek a decomposition of the
Markov structure into segregated functional networks using decomposable graphs:
a set of strongly-connected and partially overlapping cliques. We introduce a
new method to efficiently extract such cliques on a large, strongly-connected
graph. We compare methods learning different graph structures from functional
connectivity by testing the goodness of fit of the model they learn on new
data. We find that summarizing the structure as strongly-connected networks can
give a good description only for very large and overlapping networks. These
results highlight that Markov models are good tools to identify the structure
of brain connectivity from fMRI signals, but for this purpose they must reflect
the small-world properties of the underlying neural systems
Algebraic Aspects of Conditional Independence and Graphical Models
This chapter of the forthcoming Handbook of Graphical Models contains an
overview of basic theorems and techniques from algebraic geometry and how they
can be applied to the study of conditional independence and graphical models.
It also introduces binomial ideals and some ideas from real algebraic geometry.
When random variables are discrete or Gaussian, tools from computational
algebraic geometry can be used to understand implications between conditional
independence statements. This is accomplished by computing primary
decompositions of conditional independence ideals. As examples the chapter
presents in detail the graphical model of a four cycle and the intersection
axiom, a certain implication of conditional independence statements. Another
important problem in the area is to determine all constraints on a graphical
model, for example, equations determined by trek separation. The full set of
equality constraints can be determined by computing the model's vanishing
ideal. The chapter illustrates these techniques and ideas with examples from
the literature and provides references for further reading.Comment: 20 pages, 1 figur
Conjunctive Bayesian networks
Conjunctive Bayesian networks (CBNs) are graphical models that describe the
accumulation of events which are constrained in the order of their occurrence.
A CBN is given by a partial order on a (finite) set of events. CBNs generalize
the oncogenetic tree models of Desper et al. by allowing the occurrence of an
event to depend on more than one predecessor event. The present paper studies
the statistical and algebraic properties of CBNs. We determine the maximum
likelihood parameters and present a combinatorial solution to the model
selection problem. Our method performs well on two datasets where the events
are HIV mutations associated with drug resistance. Concluding with a study of
the algebraic properties of CBNs, we show that CBNs are toric varieties after a
coordinate transformation and that their ideals possess a quadratic Gr\"{o}bner
basis.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6133 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Binary Models for Marginal Independence
Log-linear models are a classical tool for the analysis of contingency
tables. In particular, the subclass of graphical log-linear models provides a
general framework for modelling conditional independences. However, with the
exception of special structures, marginal independence hypotheses cannot be
accommodated by these traditional models. Focusing on binary variables, we
present a model class that provides a framework for modelling marginal
independences in contingency tables. The approach taken is graphical and draws
on analogies to multivariate Gaussian models for marginal independence. For the
graphical model representation we use bi-directed graphs, which are in the
tradition of path diagrams. We show how the models can be parameterized in a
simple fashion, and how maximum likelihood estimation can be performed using a
version of the Iterated Conditional Fitting algorithm. Finally we consider
combining these models with symmetry restrictions
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