315 research outputs found
Automated Analytic Asymptotic Evaluation of the Marginal Likelihood for Latent Models
We present and implement two algorithms for analytic asymptotic evaluation of
the marginal likelihood of data given a Bayesian network with hidden nodes. As
shown by previous work, this evaluation is particularly hard for latent
Bayesian network models, namely networks that include hidden variables, where
asymptotic approximation deviates from the standard BIC score. Our algorithms
solve two central difficulties in asymptotic evaluation of marginal likelihood
integrals, namely, evaluation of regular dimensionality drop for latent
Bayesian network models and computation of non-standard approximation formulas
for singular statistics for these models. The presented algorithms are
implemented in Matlab and Maple and their usage is demonstrated for marginal
likelihood approximations for Bayesian networks with hidden variables.Comment: Appears in Proceedings of the Nineteenth Conference on Uncertainty in
Artificial Intelligence (UAI2003
On the Logic of Causal Models
This paper explores the role of Directed Acyclic Graphs (DAGs) as a
representation of conditional independence relationships. We show that DAGs
offer polynomially sound and complete inference mechanisms for inferring
conditional independence relationships from a given causal set of such
relationships. As a consequence, d-separation, a graphical criterion for
identifying independencies in a DAG, is shown to uncover more valid
independencies then any other criterion. In addition, we employ the Armstrong
property of conditional independence to show that the dependence relationships
displayed by a DAG are inherently consistent, i.e. for every DAG D there exists
some probability distribution P that embodies all the conditional
independencies displayed in D and none other.Comment: Appears in Proceedings of the Fourth Conference on Uncertainty in
Artificial Intelligence (UAI1988
Separable and transitive graphoids
We examine three probabilistic formulations of the sentence a and b are
totally unrelated with respect to a given set of variables U. First, two
variables a and b are totally independent if they are independent given any
value of any subset of the variables in U. Second, two variables are totally
uncoupled if U can be partitioned into two marginally independent sets
containing a and b respectively. Third, two variables are totally disconnected
if the corresponding nodes are disconnected in every belief network
representation. We explore the relationship between these three formulations of
unrelatedness and explain their relevance to the process of acquiring
probabilistic knowledge from human experts.Comment: Appears in Proceedings of the Sixth Conference on Uncertainty in
Artificial Intelligence (UAI1990
Importance Sampling via Variational Optimization
Computing the exact likelihood of data in large Bayesian networks consisting
of thousands of vertices is often a difficult task. When these models contain
many deterministic conditional probability tables and when the observed values
are extremely unlikely even alternative algorithms such as variational methods
and stochastic sampling often perform poorly. We present a new importance
sampling algorithm for Bayesian networks which is based on variational
techniques. We use the updates of the importance function to predict whether
the stochastic sampling converged above or below the true likelihood, and
change the proposal distribution accordingly. The validity of the method and
its contribution to convergence is demonstrated on hard networks of large
genetic linkage analysis tasks.Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty
in Artificial Intelligence (UAI2007
Graphical Models and Exponential Families
We provide a classification of graphical models according to their
representation as subfamilies of exponential families. Undirected graphical
models with no hidden variables are linear exponential families (LEFs),
directed acyclic graphical models and chain graphs with no hidden variables,
including Bayesian networks with several families of local distributions, are
curved exponential families (CEFs) and graphical models with hidden variables
are stratified exponential families (SEFs). An SEF is a finite union of CEFs
satisfying a frontier condition. In addition, we illustrate how one can
automatically generate independence and non-independence constraints on the
distributions over the observable variables implied by a Bayesian network with
hidden variables. The relevance of these results for model selection is
examined.Comment: Appears in Proceedings of the Fourteenth Conference on Uncertainty in
Artificial Intelligence (UAI1998
Approximation Algorithms for the Loop Cutset Problem
We show how to find a small loop curser in a Bayesian network. Finding such a
loop cutset is the first step in the method of conditioning for inference. Our
algorithm for finding a loop cutset, called MGA, finds a loop cutset which is
guaranteed in the worst case to contain less than twice the number of variables
contained in a minimum loop cutset. We test MGA on randomly generated graphs
and find that the average ratio between the number of instances associated with
the algorithms' output and the number of instances associated with a minimum
solution is 1.22.Comment: Appears in Proceedings of the Tenth Conference on Uncertainty in
Artificial Intelligence (UAI1994
Inference Algorithms for Similarity Networks
We examine two types of similarity networks each based on a distinct notion
of relevance. For both types of similarity networks we present an efficient
inference algorithm that works under the assumption that every event has a
nonzero probability of occurrence. Another inference algorithm is developed for
type 1 similarity networks that works under no restriction, albeit less
efficiently.Comment: Appears in Proceedings of the Ninth Conference on Uncertainty in
Artificial Intelligence (UAI1993
Learning Bayesian Networks: A Unification for Discrete and Gaussian Domains
We examine Bayesian methods for learning Bayesian networks from a combination
of prior knowledge and statistical data. In particular, we unify the approaches
we presented at last year's conference for discrete and Gaussian domains. We
derive a general Bayesian scoring metric, appropriate for both domains. We then
use this metric in combination with well-known statistical facts about the
Dirichlet and normal--Wishart distributions to derive our metrics for discrete
and Gaussian domains.Comment: Appears in Proceedings of the Eleventh Conference on Uncertainty in
Artificial Intelligence (UAI1995
Learning Gaussian Networks
We describe algorithms for learning Bayesian networks from a combination of
user knowledge and statistical data. The algorithms have two components: a
scoring metric and a search procedure. The scoring metric takes a network
structure, statistical data, and a user's prior knowledge, and returns a score
proportional to the posterior probability of the network structure given the
data. The search procedure generates networks for evaluation by the scoring
metric. Previous work has concentrated on metrics for domains containing only
discrete variables, under the assumption that data represents a multinomial
sample. In this paper, we extend this work, developing scoring metrics for
domains containing all continuous variables or a mixture of discrete and
continuous variables, under the assumption that continuous data is sampled from
a multivariate normal distribution. Our work extends traditional statistical
approaches for identifying vanishing regression coefficients in that we
identify two important assumptions, called event equivalence and parameter
modularity, that when combined allow the construction of prior distributions
for multivariate normal parameters from a single prior Bayesian network
specified by a user.Comment: Appears in Proceedings of the Tenth Conference on Uncertainty in
Artificial Intelligence (UAI1994
Asymptotic Model Selection for Directed Networks with Hidden Variables
We extend the Bayesian Information Criterion (BIC), an asymptotic
approximation for the marginal likelihood, to Bayesian networks with hidden
variables. This approximation can be used to select models given large samples
of data. The standard BIC as well as our extension punishes the complexity of a
model according to the dimension of its parameters. We argue that the dimension
of a Bayesian network with hidden variables is the rank of the Jacobian matrix
of the transformation between the parameters of the network and the parameters
of the observable variables. We compute the dimensions of several networks
including the naive Bayes model with a hidden root node.Comment: Appears in Proceedings of the Twelfth Conference on Uncertainty in
Artificial Intelligence (UAI1996
- …