509 research outputs found
Markov Properties for Graphical Models with Cycles and Latent Variables
We investigate probabilistic graphical models that allow for both cycles and
latent variables. For this we introduce directed graphs with hyperedges
(HEDGes), generalizing and combining both marginalized directed acyclic graphs
(mDAGs) that can model latent (dependent) variables, and directed mixed graphs
(DMGs) that can model cycles. We define and analyse several different Markov
properties that relate the graphical structure of a HEDG with a probability
distribution on a corresponding product space over the set of nodes, for
example factorization properties, structural equations properties,
ordered/local/global Markov properties, and marginal versions of these. The
various Markov properties for HEDGes are in general not equivalent to each
other when cycles or hyperedges are present, in contrast with the simpler case
of directed acyclic graphical (DAG) models (also known as Bayesian networks).
We show how the Markov properties for HEDGes - and thus the corresponding
graphical Markov models - are logically related to each other.Comment: 131 page
Marginal AMP Chain Graphs
We present a new family of models that is based on graphs that may have
undirected, directed and bidirected edges. We name these new models marginal
AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP
chain graph under marginalization of some of its nodes. However, MAMP chain
graphs do not only subsume AMP chain graphs but also multivariate regression
chain graphs. We describe global and pairwise Markov properties for MAMP chain
graphs and prove their equivalence for compositional graphoids. We also
characterize when two MAMP chain graphs are Markov equivalent.
For Gaussian probability distributions, we also show that every MAMP chain
graph is Markov equivalent to some directed and acyclic graph with
deterministic nodes under marginalization and conditioning on some of its
nodes. This is important because it implies that the independence model
represented by a MAMP chain graph can be accounted for by some data generating
process that is partially observed and has selection bias. Finally, we modify
MAMP chain graphs so that they are closed under marginalization for Gaussian
probability distributions. This is a desirable feature because it guarantees
parsimonious models under marginalization.Comment: Changes from v1 to v2: Discussion section got extended. Changes from
v2 to v3: New Sections 3 and 5. Changes from v3 to v4: Example 4 added to
discussion section. Changes from v4 to v5: None. Changes from v5 to v6: Some
minor and major errors have been corrected. The latter include the
definitions of descending route and pairwise separation base, and the proofs
of Theorems 5 and
The Inflation Technique for Causal Inference with Latent Variables
The problem of causal inference is to determine if a given probability
distribution on observed variables is compatible with some causal structure.
The difficult case is when the causal structure includes latent variables. We
here introduce the for tackling this problem. An
inflation of a causal structure is a new causal structure that can contain
multiple copies of each of the original variables, but where the ancestry of
each copy mirrors that of the original. To every distribution of the observed
variables that is compatible with the original causal structure, we assign a
family of marginal distributions on certain subsets of the copies that are
compatible with the inflated causal structure. It follows that compatibility
constraints for the inflation can be translated into compatibility constraints
for the original causal structure. Even if the constraints at the level of
inflation are weak, such as observable statistical independences implied by
disjoint causal ancestry, the translated constraints can be strong. We apply
this method to derive new inequalities whose violation by a distribution
witnesses that distribution's incompatibility with the causal structure (of
which Bell inequalities and Pearl's instrumental inequality are prominent
examples). We describe an algorithm for deriving all such inequalities for the
original causal structure that follow from ancestral independences in the
inflation. For three observed binary variables with pairwise common causes, it
yields inequalities that are stronger in at least some aspects than those
obtainable by existing methods. We also describe an algorithm that derives a
weaker set of inequalities but is more efficient. Finally, we discuss which
inflations are such that the inequalities one obtains from them remain valid
even for quantum (and post-quantum) generalizations of the notion of a causal
model.Comment: Minor final corrections, updated to match the published version as
closely as possibl
Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study
This paper proposes a method for construction of approximate feasible primal
solutions from dual ones for large-scale optimization problems possessing
certain separability properties. Whereas infeasible primal estimates can
typically be produced from (sub-)gradients of the dual function, it is often
not easy to project them to the primal feasible set, since the projection
itself has a complexity comparable to the complexity of the initial problem. We
propose an alternative efficient method to obtain feasibility and show that its
properties influencing the convergence to the optimum are similar to the
properties of the Euclidean projection. We apply our method to the local
polytope relaxation of inference problems for Markov Random Fields and
demonstrate its superiority over existing methods.Comment: 20 page, 4 figure
- …