13,424 research outputs found
Markov equivalence of marginalized local independence graphs
Symmetric independence relations are often studied using graphical
representations. Ancestral graphs or acyclic directed mixed graphs with
-separation provide classes of symmetric graphical independence models that
are closed under marginalization. Asymmetric independence relations appear
naturally for multivariate stochastic processes, for instance in terms of local
independence. However, no class of graphs representing such asymmetric
independence relations, which is also closed under marginalization, has been
developed. We develop the theory of directed mixed graphs with -separation
and show that this provides a graphical independence model class which is
closed under marginalization and which generalizes previously considered
graphical representations of local independence.
For statistical applications, it is pivotal to characterize graphs that
induce the same independence relations as such a Markov equivalence class of
graphs is the object that is ultimately identifiable from observational data.
Our main result is that for directed mixed graphs with -separation each
Markov equivalence class contains a maximal element which can be constructed
from the independence relations alone. Moreover, we introduce the directed
mixed equivalence graph as the maximal graph with edge markings. This graph
encodes all the information about the edges that is identifiable from the
independence relations, and furthermore it can be computed efficiently from the
maximal graph.Comment: 49 pages (including supplementary material), updated to add examples
and fix typo
Graphical models for marked point processes based on local independence
A new class of graphical models capturing the dependence structure of events
that occur in time is proposed. The graphs represent so-called local
independences, meaning that the intensities of certain types of events are
independent of some (but not necessarily all) events in the past. This dynamic
concept of independence is asymmetric, similar to Granger non-causality, so
that the corresponding local independence graphs differ considerably from
classical graphical models. Hence a new notion of graph separation, called
delta-separation, is introduced and implications for the underlying model as
well as for likelihood inference are explored. Benefits regarding facilitation
of reasoning about and understanding of dynamic dependencies as well as
computational simplifications are discussed.Comment: To appear in the Journal of the Royal Statistical Society Series
Credal Networks under Epistemic Irrelevance
A credal network under epistemic irrelevance is a generalised type of
Bayesian network that relaxes its two main building blocks. On the one hand,
the local probabilities are allowed to be partially specified. On the other
hand, the assessments of independence do not have to hold exactly.
Conceptually, these two features turn credal networks under epistemic
irrelevance into a powerful alternative to Bayesian networks, offering a more
flexible approach to graph-based multivariate uncertainty modelling. However,
in practice, they have long been perceived as very hard to work with, both
theoretically and computationally.
The aim of this paper is to demonstrate that this perception is no longer
justified. We provide a general introduction to credal networks under epistemic
irrelevance, give an overview of the state of the art, and present several new
theoretical results. Most importantly, we explain how these results can be
combined to allow for the design of recursive inference methods. We provide
numerous concrete examples of how this can be achieved, and use these to
demonstrate that computing with credal networks under epistemic irrelevance is
most definitely feasible, and in some cases even highly efficient. We also
discuss several philosophical aspects, including the lack of symmetry, how to
deal with probability zero, the interpretation of lower expectations, the
axiomatic status of graphoid properties, and the difference between updating
and conditioning
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
Causal inference using the algorithmic Markov condition
Inferring the causal structure that links n observables is usually based upon
detecting statistical dependences and choosing simple graphs that make the
joint measure Markovian. Here we argue why causal inference is also possible
when only single observations are present.
We develop a theory how to generate causal graphs explaining similarities
between single objects. To this end, we replace the notion of conditional
stochastic independence in the causal Markov condition with the vanishing of
conditional algorithmic mutual information and describe the corresponding
causal inference rules.
We explain why a consistent reformulation of causal inference in terms of
algorithmic complexity implies a new inference principle that takes into
account also the complexity of conditional probability densities, making it
possible to select among Markov equivalent causal graphs. This insight provides
a theoretical foundation of a heuristic principle proposed in earlier work.
We also discuss how to replace Kolmogorov complexity with decidable
complexity criteria. This can be seen as an algorithmic analog of replacing the
empirically undecidable question of statistical independence with practical
independence tests that are based on implicit or explicit assumptions on the
underlying distribution.Comment: 16 figure
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