6,039 research outputs found
Hierarchical Models as Marginals of Hierarchical Models
We investigate the representation of hierarchical models in terms of
marginals of other hierarchical models with smaller interactions. We focus on
binary variables and marginals of pairwise interaction models whose hidden
variables are conditionally independent given the visible variables. In this
case the problem is equivalent to the representation of linear subspaces of
polynomials by feedforward neural networks with soft-plus computational units.
We show that every hidden variable can freely model multiple interactions among
the visible variables, which allows us to generalize and improve previous
results. In particular, we show that a restricted Boltzmann machine with less
than hidden binary variables can approximate
every distribution of visible binary variables arbitrarily well, compared
to from the best previously known result.Comment: 18 pages, 4 figures, 2 tables, WUPES'1
Positive margins and primary decomposition
We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the N-cycle and the complete bipartite graph K2,N2−2, with various restrictions on the size of the nodes
On the half-Cauchy prior for a global scale parameter
This paper argues that the half-Cauchy distribution should replace the
inverse-Gamma distribution as a default prior for a top-level scale parameter
in Bayesian hierarchical models, at least for cases where a proper prior is
necessary. Our arguments involve a blend of Bayesian and frequentist reasoning,
and are intended to complement the original case made by Gelman (2006) in
support of the folded-t family of priors. First, we generalize the half-Cauchy
prior to the wider class of hypergeometric inverted-beta priors. We derive
expressions for posterior moments and marginal densities when these priors are
used for a top-level normal variance in a Bayesian hierarchical model. We go on
to prove a proposition that, together with the results for moments and
marginals, allows us to characterize the frequentist risk of the Bayes
estimators under all global-shrinkage priors in the class. These theoretical
results, in turn, allow us to study the frequentist properties of the
half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies
a sensible 'middle ground' within this class: it performs very well near the
origin, but does not lead to drastic compromises in other parts of the
parameter space. This provides an alternative, classical justification for the
repeated, routine use of this prior. We also consider situations where the
underlying mean vector is sparse, where we argue that the usual conjugate
choice of an inverse-gamma prior is particularly inappropriate, and can lead to
highly distorted posterior inferences. Finally, we briefly summarize some open
issues in the specification of default priors for scale terms in hierarchical
models
Positive margins and primary decomposition
We study random walks on contingency tables with fixed marginals,
corresponding to a (log-linear) hierarchical model. If the set of allowed moves
is not a Markov basis, then there exist tables with the same marginals that are
not connected. We study linear conditions on the values of the marginals that
ensure that all tables in a given fiber are connected. We show that many
graphical models have the positive margins property, which says that all fibers
with strictly positive marginals are connected by the quadratic moves that
correspond to conditional independence statements. The property persists under
natural operations such as gluing along cliques, but we also construct examples
of graphical models not enjoying this property. We also provide a negative
answer to a question of Engstr\"om, Kahle, and Sullivant by demonstrating that
the global Markov ideal of the complete bipartite graph K_(3,3) is not radical.
Our analysis of the positive margins property depends on computing the
primary decomposition of the associated conditional independence ideal. The
main technical results of the paper are primary decompositions of the
conditional independence ideals of graphical models of the -cycle and the
complete bipartite graph , with various restrictions on the size of
the nodes.Comment: 26 pages, 3 figures, v2: various small improvements, v3: added
K_(3,3) as an example of a non-radical global Markov ideal + small
improvement
Conditions for swappability of records in a microdata set when some marginals are fixed
We consider swapping of two records in a microdata set for the purpose of
disclosure control. We give some necessary and sufficient conditions that some
observations can be swapped between two records under the restriction that a
given set of marginals are fixed. We also give an algorithm to find another
record for swapping if one wants to swap out some observations from a
particular record. Our result has a close connection to the construction of
Markov bases for contingency tables with given marginals
Infinite factorization of multiple non-parametric views
Combined analysis of multiple data sources has increasing application interest, in particular for distinguishing shared and source-specific aspects. We extend this rationale of classical canonical correlation analysis into a flexible, generative and non-parametric clustering
setting, by introducing a novel non-parametric hierarchical
mixture model. The lower level of the model describes each source with a flexible non-parametric mixture, and the top level combines these to describe commonalities of the sources. The lower-level clusters arise from hierarchical Dirichlet Processes, inducing an infinite-dimensional contingency table between the views. The commonalities between the sources are modeled by an infinite block
model of the contingency table, interpretable as non-negative factorization of infinite matrices, or as a prior for infinite contingency tables. With Gaussian mixture components plugged in for continuous measurements, the model is applied to two views of genes, mRNA expression and abundance of the produced proteins, to expose groups of genes that are co-regulated in either or both of the views.
Cluster analysis of co-expression is a standard simple way of screening for co-regulation, and the two-view analysis extends the approach to distinguishing between pre- and post-translational regulation
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