37,015 research outputs found
Incomplete graphical model inference via latent tree aggregation
Graphical network inference is used in many fields such as genomics or
ecology to infer the conditional independence structure between variables, from
measurements of gene expression or species abundances for instance. In many
practical cases, not all variables involved in the network have been observed,
and the samples are actually drawn from a distribution where some variables
have been marginalized out. This challenges the sparsity assumption commonly
made in graphical model inference, since marginalization yields locally dense
structures, even when the original network is sparse. We present a procedure
for inferring Gaussian graphical models when some variables are unobserved,
that accounts both for the influence of missing variables and the low density
of the original network. Our model is based on the aggregation of spanning
trees, and the estimation procedure on the Expectation-Maximization algorithm.
We treat the graph structure and the unobserved nodes as missing variables and
compute posterior probabilities of edge appearance. To provide a complete
methodology, we also propose several model selection criteria to estimate the
number of missing nodes. A simulation study and an illustration flow cytometry
data reveal that our method has favorable edge detection properties compared to
existing graph inference techniques. The methods are implemented in an R
package
Credal Networks under Epistemic Irrelevance
A credal network under epistemic irrelevance is a generalised type of
Bayesian network that relaxes its two main building blocks. On the one hand,
the local probabilities are allowed to be partially specified. On the other
hand, the assessments of independence do not have to hold exactly.
Conceptually, these two features turn credal networks under epistemic
irrelevance into a powerful alternative to Bayesian networks, offering a more
flexible approach to graph-based multivariate uncertainty modelling. However,
in practice, they have long been perceived as very hard to work with, both
theoretically and computationally.
The aim of this paper is to demonstrate that this perception is no longer
justified. We provide a general introduction to credal networks under epistemic
irrelevance, give an overview of the state of the art, and present several new
theoretical results. Most importantly, we explain how these results can be
combined to allow for the design of recursive inference methods. We provide
numerous concrete examples of how this can be achieved, and use these to
demonstrate that computing with credal networks under epistemic irrelevance is
most definitely feasible, and in some cases even highly efficient. We also
discuss several philosophical aspects, including the lack of symmetry, how to
deal with probability zero, the interpretation of lower expectations, the
axiomatic status of graphoid properties, and the difference between updating
and conditioning
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
Identifiability of Causal Graphs using Functional Models
This work addresses the following question: Under what assumptions on the
data generating process can one infer the causal graph from the joint
distribution? The approach taken by conditional independence-based causal
discovery methods is based on two assumptions: the Markov condition and
faithfulness. It has been shown that under these assumptions the causal graph
can be identified up to Markov equivalence (some arrows remain undirected)
using methods like the PC algorithm. In this work we propose an alternative by
defining Identifiable Functional Model Classes (IFMOCs). As our main theorem we
prove that if the data generating process belongs to an IFMOC, one can identify
the complete causal graph. To the best of our knowledge this is the first
identifiability result of this kind that is not limited to linear functional
relationships. We discuss how the IFMOC assumption and the Markov and
faithfulness assumptions relate to each other and explain why we believe that
the IFMOC assumption can be tested more easily on given data. We further
provide a practical algorithm that recovers the causal graph from finitely many
data; experiments on simulated data support the theoretical findings
Inferring dynamic genetic networks with low order independencies
In this paper, we propose a novel inference method for dynamic genetic
networks which makes it possible to face with a number of time measurements n
much smaller than the number of genes p. The approach is based on the concept
of low order conditional dependence graph that we extend here in the case of
Dynamic Bayesian Networks. Most of our results are based on the theory of
graphical models associated with the Directed Acyclic Graphs (DAGs). In this
way, we define a minimal DAG G which describes exactly the full order
conditional dependencies given the past of the process. Then, to face with the
large p and small n estimation case, we propose to approximate DAG G by
considering low order conditional independencies. We introduce partial qth
order conditional dependence DAGs G(q) and analyze their probabilistic
properties. In general, DAGs G(q) differ from DAG G but still reflect relevant
dependence facts for sparse networks such as genetic networks. By using this
approximation, we set out a non-bayesian inference method and demonstrate the
effectiveness of this approach on both simulated and real data analysis. The
inference procedure is implemented in the R package 'G1DBN' freely available
from the CRAN archive
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