317 research outputs found
Equivalence Classes of Staged Trees
In this paper we give a complete characterization of the statistical
equivalence classes of CEGs and of staged trees. We are able to show that all
graphical representations of the same model share a common polynomial
description. Then, simple transformations on that polynomial enable us to
traverse the corresponding class of graphs. We illustrate our results with a
real analysis of the implicit dependence relationships within a previously
studied dataset.Comment: 18 pages, 4 figure
Tree cumulants and the geometry of binary tree models
In this paper we investigate undirected discrete graphical tree models when
all the variables in the system are binary, where leaves represent the
observable variables and where all the inner nodes are unobserved. A novel
approach based on the theory of partially ordered sets allows us to obtain a
convenient parametrization of this model class. The construction of the
proposed coordinate system mirrors the combinatorial definition of cumulants. A
simple product-like form of the resulting parametrization gives insight into
identifiability issues associated with this model class. In particular, we
provide necessary and sufficient conditions for such a model to be identified
up to the switching of labels of the inner nodes. When these conditions hold,
we give explicit formulas for the parameters of the model. Whenever the model
fails to be identified, we use the new parametrization to describe the geometry
of the unidentified parameter space. We illustrate these results using a simple
example.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ338 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The Dependence of Routine Bayesian Model Selection Methods on Irrelevant Alternatives
Bayesian methods - either based on Bayes Factors or BIC - are now widely used
for model selection. One property that might reasonably be demanded of any
model selection method is that if a model is preferred to a model
, when these two models are expressed as members of one model class
, this preference is preserved when they are embedded in a
different class . However, we illustrate in this paper that with
the usual implementation of these common Bayesian procedures this property does
not hold true even approximately. We therefore contend that to use these
methods it is first necessary for there to exist a "natural" embedding class.
We argue that in any context like the one illustrated in our running example of
Bayesian model selection of binary phylogenetic trees there is no such
embedding
Conditionally externally Bayesian pooling operators in chain graphs
We address the multivariate version of French’s group decision problem where the m members of a group, who are jointly responsible for the decisions they should make, wish to combine their beliefs about the possible values of n random variables into the group consensus probability distribution. We shall assume the group has agreed on the structure of associations of variables in a problem, as might be represented by a commonly agreed partially complete chain graph (PCG) we define in the paper. However, the members diverge about the actual conditional probability distributions for the variables in the common PCG. The combination algorithm we suggest they adopt is one which demands, at least on learning information which is common to the members and which preserves the originally agreed PCG structure, that the pools of conditional
distributions associated with the PCG are externally Bayesian (EB). We propose a characterization for such conditionally EB (CEB) poolings which is more general and flexible than the characterization proposed by Genest, McConway and Schervish. In particular, such a generalization allows the weights attributed to the joint probability assessments of different individuals in the pool to differ across the distinct components of each joint density. We show that the group’s commitment to being CEB on chain elements can be accomplished by the group being EB on the whole PCG
when the group also agrees to perform the conditional poolings in an ordering compatible with evidence propagation in the graph
Directed expected utility networks
A variety of statistical graphical models have been defined to represent the conditional independences underlying a random vector of interest. Similarly, many different graphs embedding various types of preferential independences, such as, for example, conditional utility independence and generalized additive independence, have more recently started to appear. In this paper, we define a new graphical model, called a directed expected utility network, whose edges depict both probabilistic and utility conditional independences. These embed a very flexible class of utility models, much larger than those usually conceived in standard influence diagrams. Our graphical representation and various transformations of the original graph into a tree structure are then used to guide fast routines for the computation of a decision problem’s expected utilities. We show that our routines generalize those usually utilized in standard influence diagrams’ evaluations under much more restrictive conditions. We then proceed with the construction of a directed expected utility network to support decision makers in the domain of household food security
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