10,490 research outputs found
On the geometry of Bayesian graphical models with hidden variables
Abstract In this paper we investigate the geometry of the likelihood of the unknown parameters in a simple class of Bayesian directed graphs with hidden variables. This enables us, before any numerical algorithms are employed, to obtain certain insights in the nature of the uniden tifiability inherent in such models, the way posterior densities will be sensitive to prior densities and the typical geometrical form these posterior densities might take. Many of these insights carry over into more com plicated Bayesian networks with systematic missing data
Parametric Inference for Biological Sequence Analysis
One of the major successes in computational biology has been the unification,
using the graphical model formalism, of a multitude of algorithms for
annotating and comparing biological sequences. Graphical models that have been
applied towards these problems include hidden Markov models for annotation,
tree models for phylogenetics, and pair hidden Markov models for alignment. A
single algorithm, the sum-product algorithm, solves many of the inference
problems associated with different statistical models. This paper introduces
the \emph{polytope propagation algorithm} for computing the Newton polytope of
an observation from a graphical model. This algorithm is a geometric version of
the sum-product algorithm and is used to analyze the parametric behavior of
maximum a posteriori inference calculations for graphical models.Comment: 15 pages, 4 figures. See also companion paper "Tropical Geometry of
Statistical Models" (q-bio.QM/0311009
Algebraic geometry of Gaussian Bayesian networks
Conditional independence models in the Gaussian case are algebraic varieties
in the cone of positive definite covariance matrices. We study these varieties
in the case of Bayesian networks, with a view towards generalizing the
recursive factorization theorem to situations with hidden variables. In the
case when the underlying graph is a tree, we show that the vanishing ideal of
the model is generated by the conditional independence statements implied by
graph. We also show that the ideal of any Bayesian network is homogeneous with
respect to a multigrading induced by a collection of upstream random variables.
This has a number of important consequences for hidden variable models.
Finally, we relate the ideals of Bayesian networks to a number of classical
constructions in algebraic geometry including toric degenerations of the
Grassmannian, matrix Schubert varieties, and secant varieties.Comment: 30 page, 4 figure
Tropical Geometry of Statistical Models
This paper presents a unified mathematical framework for inference in
graphical models, building on the observation that graphical models are
algebraic varieties.
From this geometric viewpoint, observations generated from a model are
coordinates of a point in the variety, and the sum-product algorithm is an
efficient tool for evaluating specific coordinates. The question addressed here
is how the solutions to various inference problems depend on the model
parameters. The proposed answer is expressed in terms of tropical algebraic
geometry. A key role is played by the Newton polytope of a statistical model.
Our results are applied to the hidden Markov model and to the general Markov
model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion
paper, "Parametric Inference for Biological Sequence Analysis
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