125,712 research outputs found
Graphs for margins of Bayesian networks
Directed acyclic graph (DAG) models, also called Bayesian networks, impose
conditional independence constraints on a multivariate probability
distribution, and are widely used in probabilistic reasoning, machine learning
and causal inference. If latent variables are included in such a model, then
the set of possible marginal distributions over the remaining (observed)
variables is generally complex, and not represented by any DAG. Larger classes
of mixed graphical models, which use multiple edge types, have been introduced
to overcome this; however, these classes do not represent all the models which
can arise as margins of DAGs. In this paper we show that this is because
ordinary mixed graphs are fundamentally insufficiently rich to capture the
variety of marginal models.
We introduce a new class of hyper-graphs, called mDAGs, and a latent
projection operation to obtain an mDAG from the margin of a DAG. We show that
each distinct marginal of a DAG model is represented by at least one mDAG, and
provide graphical results towards characterizing when two such marginal models
are the same. Finally we show that mDAGs correctly capture the marginal
structure of causally-interpreted DAGs under interventions on the observed
variables
Neural Graphical Models
Probabilistic Graphical Models are often used to understand dynamics of a
system. They can model relationships between features (nodes) and the
underlying distribution. Theoretically these models can represent very complex
dependency functions, but in practice often simplifying assumptions are made
due to computational limitations associated with graph operations. In this work
we introduce Neural Graphical Models (NGMs) which attempt to represent complex
feature dependencies with reasonable computational costs. Given a graph of
feature relationships and corresponding samples, we capture the dependency
structure between the features along with their complex function
representations by using a neural network as a multi-task learning framework.
We provide efficient learning, inference and sampling algorithms. NGMs can fit
generic graph structures including directed, undirected and mixed-edge graphs
as well as support mixed input data types. We present empirical studies that
show NGMs' capability to represent Gaussian graphical models, perform inference
analysis of a lung cancer data and extract insights from a real world infant
mortality data provided by Centers for Disease Control and Prevention
Spectral Methods for Learning Multivariate Latent Tree Structure
This work considers the problem of learning the structure of multivariate
linear tree models, which include a variety of directed tree graphical models
with continuous, discrete, and mixed latent variables such as linear-Gaussian
models, hidden Markov models, Gaussian mixture models, and Markov evolutionary
trees. The setting is one where we only have samples from certain observed
variables in the tree, and our goal is to estimate the tree structure (i.e.,
the graph of how the underlying hidden variables are connected to each other
and to the observed variables). We propose the Spectral Recursive Grouping
algorithm, an efficient and simple bottom-up procedure for recovering the tree
structure from independent samples of the observed variables. Our finite sample
size bounds for exact recovery of the tree structure reveal certain natural
dependencies on underlying statistical and structural properties of the
underlying joint distribution. Furthermore, our sample complexity guarantees
have no explicit dependence on the dimensionality of the observed variables,
making the algorithm applicable to many high-dimensional settings. At the heart
of our algorithm is a spectral quartet test for determining the relative
topology of a quartet of variables from second-order statistics
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