24 research outputs found
Exploiting Contextual Independence In Probabilistic Inference
Bayesian belief networks have grown to prominence because they provide
compact representations for many problems for which probabilistic inference is
appropriate, and there are algorithms to exploit this compactness. The next
step is to allow compact representations of the conditional probabilities of a
variable given its parents. In this paper we present such a representation that
exploits contextual independence in terms of parent contexts; which variables
act as parents may depend on the value of other variables. The internal
representation is in terms of contextual factors (confactors) that is simply a
pair of a context and a table. The algorithm, contextual variable elimination,
is based on the standard variable elimination algorithm that eliminates the
non-query variables in turn, but when eliminating a variable, the tables that
need to be multiplied can depend on the context. This algorithm reduces to
standard variable elimination when there is no contextual independence
structure to exploit. We show how this can be much more efficient than variable
elimination when there is structure to exploit. We explain why this new method
can exploit more structure than previous methods for structured belief network
inference and an analogous algorithm that uses trees
First-Order Decomposition Trees
Lifting attempts to speed up probabilistic inference by exploiting symmetries
in the model. Exact lifted inference methods, like their propositional
counterparts, work by recursively decomposing the model and the problem. In the
propositional case, there exist formal structures, such as decomposition trees
(dtrees), that represent such a decomposition and allow us to determine the
complexity of inference a priori. However, there is currently no equivalent
structure nor analogous complexity results for lifted inference. In this paper,
we introduce FO-dtrees, which upgrade propositional dtrees to the first-order
level. We show how these trees can characterize a lifted inference solution for
a probabilistic logical model (in terms of a sequence of lifted operations),
and make a theoretical analysis of the complexity of lifted inference in terms
of the novel notion of lifted width for the tree
Modelling with non-stratified chain event graphs
© 2019, Springer Nature Switzerland AG. Chain Event Graphs (CEGs) are recent probabilistic graphical modelling tools that have proved successful in modelling scenarios with context-specific independencies. Although the theory underlying CEGs supports appropriate representation of structural zeroes, the literature so far does not provide an adaptation of the vanilla CEG methods for a real-world application presenting structural zeroes also known as the non-stratified CEG class. To illustrate these methods, we present a non-stratified CEG representing a public health intervention designed to reduce the risk and rate of falling in the elderly. We then compare the CEG model to the more conventional Bayesian Network model when applied to this setting
Refining a Bayesian network using a chain event graph
The search for a useful explanatory model based on a Bayesian Network (BN) now has a long and successful history. However, when the dependence structure between the variables of the problem is asymmetric then this cannot be captured by the BN. The Chain Event Graph (CEG) provides a richer class of models which incorporates these types of dependence structures as well as retaining the property that conclusions can be easily read back to the client. We demonstrate on a real health study how the CEG leads us to promising higher scoring models and further enables us to make more refined conclusions than can be made from the BN. Further we show how these graphs can express causal hypotheses about possible interventions that could be enforced
Labeled Directed Acyclic Graphs: a generalization of context-specific independence in directed graphical models
We introduce a novel class of labeled directed acyclic graph (LDAG) models
for finite sets of discrete variables. LDAGs generalize earlier proposals for
allowing local structures in the conditional probability distribution of a
node, such that unrestricted label sets determine which edges can be deleted
from the underlying directed acyclic graph (DAG) for a given context. Several
properties of these models are derived, including a generalization of the
concept of Markov equivalence classes. Efficient Bayesian learning of LDAGs is
enabled by introducing an LDAG-based factorization of the Dirichlet prior for
the model parameters, such that the marginal likelihood can be calculated
analytically. In addition, we develop a novel prior distribution for the model
structures that can appropriately penalize a model for its labeling complexity.
A non-reversible Markov chain Monte Carlo algorithm combined with a greedy hill
climbing approach is used for illustrating the useful properties of LDAG models
for both real and synthetic data sets.Comment: 26 pages, 17 figure
A Score-and-Search Approach to Learning Bayesian Networks with Noisy-OR Relations
A Bayesian network is a probabilistic graphical model that consists of a
directed acyclic graph (DAG), where each node is a random variable and attached
to each node is a conditional probability distribution (CPD). A Bayesian
network can be learned from data using the well-known score-and-search
approach, and within this approach a key consideration is how to simultaneously
learn the global structure in the form of the underlying DAG and the local
structure in the CPDs. Several useful forms of local structure have been
identified in the literature but thus far the score-and-search approach has
only been extended to handle local structure in form of context-specific
independence. In this paper, we show how to extend the score-and-search
approach to the important and widely useful case of noisy-OR relations. We
provide an effective gradient descent algorithm to score a candidate noisy-OR
using the widely used BIC score and we provide pruning rules that allow the
search to successfully scale to medium sized networks. Our empirical results
provide evidence for the success of our approach to learning Bayesian networks
that incorporate noisy-OR relations.Comment: Accepted to Probabilistic Graphical Models, 202