1,115 research outputs found
Bayesian clustering in decomposable graphs
In this paper we propose a class of prior distributions on decomposable
graphs, allowing for improved modeling flexibility. While existing methods
solely penalize the number of edges, the proposed work empowers practitioners
to control clustering, level of separation, and other features of the graph.
Emphasis is placed on a particular prior distribution which derives its
motivation from the class of product partition models; the properties of this
prior relative to existing priors is examined through theory and simulation. We
then demonstrate the use of graphical models in the field of agriculture,
showing how the proposed prior distribution alleviates the inflexibility of
previous approaches in properly modeling the interactions between the yield of
different crop varieties.Comment: 3 figures, 1 tabl
Sparse covariance estimation in heterogeneous samples
Standard Gaussian graphical models (GGMs) implicitly assume that the
conditional independence among variables is common to all observations in the
sample. However, in practice, observations are usually collected form
heterogeneous populations where such assumption is not satisfied, leading in
turn to nonlinear relationships among variables. To tackle these problems we
explore mixtures of GGMs; in particular, we consider both infinite mixture
models of GGMs and infinite hidden Markov models with GGM emission
distributions. Such models allow us to divide a heterogeneous population into
homogenous groups, with each cluster having its own conditional independence
structure. The main advantage of considering infinite mixtures is that they
allow us easily to estimate the number of number of subpopulations in the
sample. As an illustration, we study the trends in exchange rate fluctuations
in the pre-Euro era. This example demonstrates that the models are very
flexible while providing extremely interesting interesting insights into
real-life applications
High-dimensional learning of linear causal networks via inverse covariance estimation
We establish a new framework for statistical estimation of directed acyclic
graphs (DAGs) when data are generated from a linear, possibly non-Gaussian
structural equation model. Our framework consists of two parts: (1) inferring
the moralized graph from the support of the inverse covariance matrix; and (2)
selecting the best-scoring graph amongst DAGs that are consistent with the
moralized graph. We show that when the error variances are known or estimated
to close enough precision, the true DAG is the unique minimizer of the score
computed using the reweighted squared l_2-loss. Our population-level results
have implications for the identifiability of linear SEMs when the error
covariances are specified up to a constant multiple. On the statistical side,
we establish rigorous conditions for high-dimensional consistency of our
two-part algorithm, defined in terms of a "gap" between the true DAG and the
next best candidate. Finally, we demonstrate that dynamic programming may be
used to select the optimal DAG in linear time when the treewidth of the
moralized graph is bounded.Comment: 41 pages, 7 figure
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
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