3,072 research outputs found
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
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Standard imsets for undirected and chain graphical models
We derive standard imsets for undirected graphical models and chain graphical
models. Standard imsets for undirected graphical models are described in terms
of minimal triangulations for maximal prime subgraphs of the undirected graphs.
For describing standard imsets for chain graphical models, we first define a
triangulation of a chain graph. We then use the triangulation to generalize our
results for the undirected graphs to chain graphs.Comment: Published at http://dx.doi.org/10.3150/14-BEJ611 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Faithfulness and learning hypergraphs from discrete distributions
The concepts of faithfulness and strong-faithfulness are important for
statistical learning of graphical models. Graphs are not sufficient for
describing the association structure of a discrete distribution. Hypergraphs
representing hierarchical log-linear models are considered instead, and the
concept of parametric (strong-) faithfulness with respect to a hypergraph is
introduced. Strong-faithfulness ensures the existence of uniformly consistent
parameter estimators and enables building uniformly consistent procedures for a
hypergraph search. The strength of association in a discrete distribution can
be quantified with various measures, leading to different concepts of
strong-faithfulness. Lower and upper bounds for the proportions of
distributions that do not satisfy strong-faithfulness are computed for
different parameterizations and measures of association.Comment: 23 pages, 6 figure
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